A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. are all valid factorizations of 3.
What is the only divisor besides 1 that a prime number can have?
Ground Truth Answers: itselfitselfitselfitselfitself
Prediction:
What are numbers greater than 1 that can be divided by 3 or more numbers called?
Ground Truth Answers: composite numbercomposite numbercomposite numberprimes
Prediction:
What theorem defines the main role of primes in number theory?
Ground Truth Answers: The fundamental theorem of arithmeticfundamental theorem of arithmeticarithmeticfundamental theorem of arithmeticfundamental theorem of arithmetic
Prediction:
Any number larger than 1 can be represented as a product of what?
Ground Truth Answers: a product of primesproduct of primes that is unique up to orderingprimesprimesprimes that is unique up to ordering
Prediction:
Why must one be excluded in order to preserve the uniqueness of the fundamental theorem?
Ground Truth Answers: because one can include arbitrarily many instances of 1 in any factorizationone can include arbitrarily many instances of 1 in any factorizationcan include arbitrarily many instances of 1 in any factorizationone can include arbitrarily many instances of 1 in any factorizationbecause one can include arbitrarily many instances of 1 in any factorization
Prediction:
The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016[update], the largest known prime number has 22,338,618 decimal digits.
What is the name of the property that designates a number as being prime or not?
Ground Truth Answers: primalityprimalityprimalityprimalityprimality
Prediction:
What is the name of the process which confirms the primality of a number n?
Ground Truth Answers: trial divisiontrial divisiontrial divisiontrial divisiontrial division
Prediction:
What is the name of one algorithm useful for conveniently testing the primality of large numbers?
Ground Truth Answers: the Miller–Rabin primality testMiller–Rabin primality testMiller–Rabin primality testMiller–Rabin primality testMiller–Rabin primality test
Prediction:
What is the name of another algorithm useful for conveniently testing the primality of large numbers?
Ground Truth Answers: the AKS primality testAKS primality testAKS primality testAKS primality testAKS primality test
Prediction:
As of January 2016 how many digits does the largest known prime consist of?
Ground Truth Answers: 22,338,618 decimal digits22,338,61822,338,61822,338,61822,338,618
Prediction:
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
How many prime numbers exist?
Ground Truth Answers: infinitely manyinfinitelyinfinitely manyinfinitelyinfinitely many
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Who established the amount of prime numbers in existence?
Ground Truth Answers: EuclidEuclidEuclidEuclidEuclid
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What type of behavior in primes is it possible to determine?
Ground Truth Answers: the statistical behaviourdistributionstatisticalstatisticalstatistical
Prediction:
What theorem states that the probability that a number n is prime is inversely proportional to its logarithm?
Ground Truth Answers: the prime number theoremprime number theoremprime numberprime number theoremprime number theorem
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When was the prime number theorem proven?
Ground Truth Answers: at the end of the 19th centuryend of the 19th centuryend of the 19th centuryend of the 19th centuryend of the 19th century
Prediction:
Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.
What is the name of the supposition that any number larger than 2 can be represented as the sum of two primes?
Ground Truth Answers: Goldbach's conjectureGoldbach's conjectureGoldbach's conjectureGoldbach's conjectureGoldbach's conjecture
Prediction:
What is the name of the supposition that there are infinite pairs of primes whose difference is 2?
Ground Truth Answers: the twin prime conjecturetwin prime conjecturetwin prime conjecturetwin prime conjecturetwin prime conjecture
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Besides the analytic property of numbers, what other property of numbers does number theory focus on?
Ground Truth Answers: algebraic aspectsalgebraicalgebraicalgebraicalgebraic aspects
Prediction:
What is the application of prime numbers used in information technology which utilizes the fact that factoring very large prime numbers is very challenging?
Ground Truth Answers: public-key cryptographypublic-key cryptographypublic-key cryptographycryptographypublic-key cryptography
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What is the name of one algebraic generalization prime numbers have inspired?
Ground Truth Answers: prime idealsprime elementsprime elementsprime elements
Prediction:
Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.
Any even number larger than what cannot be considered prime?
Ground Truth Answers: 22222
Prediction:
What are the specific divisors of all even numbers larger than 2?
Ground Truth Answers: 1, 2, and n1, 2, and n1, 2, and n1, 2, and n1, 2, and n
Prediction:
What name is given to any prime number larger than 2?
Ground Truth Answers: odd primeodd primeodd primeodd primeodd prime
Prediction:
Besides 1,3 and 7, what other number must all primes greater than 5 end with?
Ground Truth Answers: 99999
Prediction:
What type of numbers are always multiples of 2?
Ground Truth Answers: even numberseveneven numberseveneven
Prediction:
Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".
What number did early Greeks not regard as a true number?
Ground Truth Answers: 11111
Prediction:
Who included 1 as the first prime number in the mid 18th century?
Ground Truth Answers: Christian GoldbachChristian GoldbachChristian Goldbachmathematiciansmathematicians
Prediction:
In the mid 18th century, who did not concur that 1 should be the first prime number?
Ground Truth Answers: Leonhard EulerLeonhard EulerLeonhard EulerLeonhard EulerLeonhard Euler
Prediction:
How many primes were included in Derrick Norman Lehmer's list of prime numbers?
Ground Truth Answers: 10,006,721primes up to 10,006,72110,006,72110,006,72110,006,721
Prediction:
What type of number do modern mathematicians consider 1 to be?
Ground Truth Answers: its own special category as a "unit"unita "unit"unita "unit
Prediction:
A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.
Which theorem would be invalid if the number 1 were considered prime?
Ground Truth Answers: Euclid's fundamental theorem of arithmeticEuclid's fundamental theorem of arithmeticarithmeticEuclid's fundamental theorem of arithmeticEuclid's fundamental theorem of arithmetic
Prediction:
The sieve of Eratosthenes would not be valid if what were true?
Ground Truth Answers: if 1 were considered a prime1 were considered a prime1 were considered a primeif 1 were considered a primeif 1 were considered a prime
Prediction:
What is another function that primes have that the number 1 does not?
Ground Truth Answers: Euler's totient functionsum of divisors functionsum of divisors functionthe sum of divisors functionsum of divisors
Prediction:
What is one function that prime numbers have that 1 does not?
Ground Truth Answers: the sum of divisors functionrelationship of the number to its corresponding value of Euler's totient functionrelationship of the number to its corresponding value of Euler's totient functionrelationship of the number to its corresponding value of Euler's totient functionthe relationship of the number to its corresponding value of Euler's totient function
Prediction:
If 1 were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers?
Ground Truth Answers: only the single number 11only the single number 1eliminate all multiples of 1eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1.
Prediction:
There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.
What is the name of the Egyptian papyrus that suggests that they may have had knowledge of prime numbers?
Ground Truth Answers: the Rhind papyrusRhindRhindEgyptian fractionRhind papyrus
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What civilization was the first known to clearly study prime numbers?
Ground Truth Answers: the Ancient GreeksAncient GreeksGreeksAncient GreeksAncient Greeks
Prediction:
What work from around 300 BC has significant theorems about prime numbers?
Ground Truth Answers: Euclid's ElementsEuclid's ElementsEuclid's ElementsEuclid's ElementsEuclid's Elements
Prediction:
Who demonstrated how to create a perfect number from a Mersenne prime?
Ground Truth Answers: EuclidEuclidEuclidEuclidEuclid
Prediction:
What does the Sieve of Eratosthenes do?
Ground Truth Answers: compute primescompute primescompute primescompute primescompute primes
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After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.
In what year did Pierre de Fermat declare Fermat's little theorem?
Ground Truth Answers: In 16401640164016401640
Prediction:
Besides Leibniz, what other mathematician proved the validity of Fermat's little theorem?
Ground Truth Answers: EulerEulerEulerEulerEuler
Prediction:
Of what form do Fermat numbers take?
Ground Truth Answers: 22n + 122n + 122n + 122n + 122n + 1
Prediction:
Of what form do Mersenne primes take?
Ground Truth Answers: 2p − 12p − 1, with p a prime2p − 12p − 12p − 1
Prediction:
To what extent did Fermat confirm the validity of Fermat numbers?
Ground Truth Answers: up to n = 4 (or 216 + 1)up to n = 4 (or 216 + 1)216 + 1n = 4n = 4
Prediction:
The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.
What is the most elemental way to test the primality of any integer n?
Ground Truth Answers: trial divisiontrial divisiontrial divisiontrial divisiontrial division
Prediction:
What makes the method of trial division more efficient?
Ground Truth Answers: if a complete list of primes up to is knowna complete list of primes up to is knowncomplete list of primes up to is knownif a complete list of primes up to is knownif a complete list of primes up to is known
Prediction:
Trial division involves dividing n by every integer m greater than what?
Ground Truth Answers: greater than 111is greater than 1 and less than or equal to the square root of n1
Prediction:
How many divisions are required to verify the primality of the number 37?
Ground Truth Answers: only three divisionsonly for those m that are primethreethreethree
Prediction:
What must the integer m be less than or equal to when performing trial division?
Ground Truth Answers: less than or equal to the square root of nthe square root of nsquare root of nthe square root of n.the square root of n.
Prediction:
Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.
How many modern types of primality tests for general numbers n are there?
Ground Truth Answers: two main classestwotwotwotwo
Prediction:
What is the name of one type of modern primality test?
Ground Truth Answers: probabilistic (or "Monte Carlo")probabilistic (or "Monte Carlo")probabilisticprobabilisticprobabilistic
Prediction:
What is the name of another type of modern primality test?
Ground Truth Answers: deterministicdeterministic algorithmsdeterministicdeterministic algorithmsdeterministic algorithms
Prediction:
What type of algorithm is trial division?
Ground Truth Answers: deterministicdeterministic algorithmdeterministicdeterministicdeterministic
Prediction:
When using a probabilistic algorithm, how is the probability that the number is composite expressed mathematically?
Ground Truth Answers: 1/(1-p)n1/(1-p)n1/(1-p)n1/(1-p)n1/(1-p)n
Prediction:
A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number.
What is one straightforward case of a probabilistic test?
Ground Truth Answers: the Fermat primality test,Fermat primality testFermat primality testFermat primality testthe Fermat primality test
Prediction:
What does the Fermat primality test depend upon?
Ground Truth Answers: np≡n (mod p)np≡n (mod p) for any n if p is a prime numbernp≡n (mod p) for any n if p is a prime numbernp≡n (mod p) for any n if p is a prime numberthe fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number
Prediction:
What type of numbers demonstrate a flaw with the Fermat primality test?
Ground Truth Answers: composite numbers (the Carmichael numbers)CarmichaelCarmichaelCarmichael numbersCarmichael numbers
Prediction:
What is the name of one impressive continuation of the Fermat primality test?
Ground Truth Answers: Baillie-PSWBaillie-PSWBaillie-PSWBaillie-PSWBaillie-PSW,
Prediction:
What is the name of another compelling continuation of the Fermat primality test?
Ground Truth Answers: Solovay-Strassen testsMiller-RabinMiller-RabinMiller-RabinMiller-Rabin
Prediction:
are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers.
Of what form are Sophie Germain primes?
Ground Truth Answers: 2p + 12p + 1 with p prime2p + 1 with p prime2p + 12p + 1
Prediction:
Of what form are Mersenne primes?
Ground Truth Answers: 2p − 12p − 12p − 1, where p is an arbitrary prime2p − 12p − 1,
Prediction:
What test is especially useful for numbers of the form 2p - 1?
Ground Truth Answers: The Lucas–Lehmer testLucas–LehmerLucas–LehmerLucas–LehmerLucas–Lehmer test
Prediction:
What is the name of one type of prime where p+1 or p-1 takes a certain shape?
Ground Truth Answers: primorial primesFermatSophie GermainSophie GermainSophie Germain
Prediction:
What is the name of another type of prime here p+1 or p-1 takes a certain shape?
Ground Truth Answers: Fermat primesMersenneprimorial primesprimorial primesprimorial primes
Prediction:
The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed]
What is the name of one type of computing method that is used to find prime numbers?
Ground Truth Answers: distributed computingdistributed computingdistributeddistributed computingdistributed computing
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In what year was the Great Internet Mersenne Prime Search project conducted?
Ground Truth Answers: In 20092009200920092009
Prediction:
the Great Internet Mersenne Prime Search, what was the prize for finding a prime with at least 10 million digits?
Ground Truth Answers: US$100,000US$100,000US$100,000$100,000US$100,000
Prediction:
What organization offers monetary awards for identifying primes with at least 100 million digits?
Ground Truth Answers: The Electronic Frontier FoundationElectronic Frontier FoundationElectronic Frontier Foundation. The Electronic Frontier Foundation$150,000
Prediction:
In what interval are some of the greatest primes without a distinct form discovered in?
Ground Truth Answers: [256kn + 1, 256k(n + 1) − 1][256kn + 1, 256k(n + 1) − 1][256kn + 1, 256k(n + 1) − 1][256kn + 1, 256k(n + 1) − 1][256kn + 1, 256k(n + 1) − 1].
Prediction:
are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.
What is name of the function used for the largest integer not greater than the number in question?
Ground Truth Answers: the floor functionfloorfloorfloor functionfloor function
Prediction:
Who first proved Bertrand's postulate?
Ground Truth Answers: ChebyshevChebyshevChebyshevChebyshevChebyshev
Prediction:
For what size natural number does Bertrand's postulate hold?
Ground Truth Answers: any natural number n > 3n > 3n > 3> 3.n > 3
Prediction:
How is the prime number p in Bertrand's postulate expressed mathematically?
Ground Truth Answers: n < p < 2n − 2n < p < 2n − 2A or μn < p < 2n − 2n < p < 2n − 2
Prediction:
On what theorem is the formula that frequently generates the number 2 and all other primes precisely once based on?
Ground Truth Answers: Wilson's theoremWilson'sWilson'sWilson's theoremWilson's theorem
Prediction:
can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.
What is another way to state the condition that infinitely many primes can exist only if a and q are coprime?
Ground Truth Answers: their greatest common divisor is onegreatest common divisor is onetheir greatest common divisor is onetheir greatest common divisor is one
Prediction:
If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of primes?
Ground Truth Answers: Dirichlet's theoremDirichlet'sDirichlet's theoremDirichlet's theorem
Prediction:
What is the density of all primes compatible with a modulo 9?
Ground Truth Answers: 1/61/61/61/6
Prediction:
If q=9 and a=3,6 or 9, how many primes would be in the progression?
Ground Truth Answers: at most one prime numberoneoneat most one
Prediction:
If q=9 and a=1,2,4,5,7, or 8, how many primes would be in a progression?
Ground Truth Answers: infinitely many prime numbersinfinitely manyinfiniteinfinitely many
Prediction:
The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,
What function is related to prime numbers?
Ground Truth Answers: The zeta functionzetazeta functionzeta function
Prediction:
What type of value would the zeta function have if there were finite primes?
Ground Truth Answers: a finite valuefinitefinitefinite
Prediction:
What property of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... shows that there is an infinite number of primes?
Ground Truth Answers: divergesdivergesexceeds any given number
Prediction:
What does it mean when a harmonic series diverges?
Ground Truth Answers: exceeds any given numberexceeds any given numberexceeds any given numberexceeds any given number
Prediction:
Of what mathematical nature is the Basel problem?
Ground Truth Answers: identityalgebraicmodern algebraic number theorymodern algebraic number theory
Prediction:
The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.
When was the Riemann hypothesis proposed?
Ground Truth Answers: 1859185918591859
Prediction:
According to the Riemann hypothesis, all zeroes of the ζ-function have real part equal to 1/2 except for what values of s?
Ground Truth Answers: s = −2, −4, ...,−2, −4, ...,−2, −4s = −2, −4
Prediction:
What does the Riemann hypothesis state the source of irregularity in the distribution of points comes from?
Ground Truth Answers: random noiserandom noiserandom noiserandom noise
Prediction:
What type of prime distribution does the Riemann hypothesis propose is also true for short intervals near X?
Ground Truth Answers: asymptotic distributionasymptoticasymptotic distributionasymptotic distribution
Prediction:
What type of prime distribution is characterized about x/log x of numbers less than x?
Ground Truth Answers: asymptotic distributionasymptoticasymptotic distributionasymptotic distribution
Prediction:
In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.
Which conjecture holds that every even integer n greater than 2 can be expressed as a sum of two primes?
Ground Truth Answers: Goldbach's conjectureGoldbach'sGoldbach'sGoldbach's
Prediction:
When did Landau propose his four conjectural problems?
Ground Truth Answers: 1912191219121912
Prediction:
As of February 2011, how many numbers has Goldbach's conjecture been proven to?
Ground Truth Answers: all numbers up to n = 2 · 1017n = 2 · 1017n = 2n = 2
Prediction:
Which theorem states that all large odd integers can be expressed as a sum of three primes?
Ground Truth Answers: Vinogradov's theoremVinogradov'sVinogradov's theoremVinogradov's theorem
Prediction:
Which theorem states that every large even integer can be written as a prime summed with a semiprime?
Ground Truth Answers: Chen's theoremChen'sChen's theoremChen's theorem
Prediction:
A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.
What conjecture holds that there is an infinite amount of twin primes?
Ground Truth Answers: twin prime conjecturetwin prime conjecturetwin prime conjecturePolignac's
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What is a twin prime?
Ground Truth Answers: pairs of primes with difference 2pairs of primes with difference 2pairs of primes with difference 2pairs of primes with difference 2
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Which conjecture holds that for any positive integer n, there is an infinite amount of pairs of consecutive primes differing by 2n?
Ground Truth Answers: Polignac's conjecturePolignac'sPolignac's conjecturePolignac's
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Of what form is the infinite amount of primes that comprise the special cases of Schinzel's hypothesis?
Ground Truth Answers: n2 + 1n2 + 1n2 + 1.n2 + 1
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What conjecture holds that there are always a minimum of 4 primes between the squares of consecutive primes greater than 2?
Ground Truth Answers: Brocard's conjectureBrocard'sBrocard's conjectureBrocard's
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For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.
Besides the study of prime numbers, what general theory was considered the official example of pure mathematics?
Ground Truth Answers: number theorynumber theorynumber theorynumber theory
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What British mathematician took pride in doing work that he felt had no military benefit?
Ground Truth Answers: G. H. HardyG. H. HardyG. H. HardyG. H. Hardy
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When was it discovered that prime numbers could applied to the creation of public key cryptography algorithms?
Ground Truth Answers: the 1970s1970s1970s1970s
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Besides public key cryptography, what is another application for prime numbers?
Ground Truth Answers: hash tableshash tableshash tables and pseudorandom number generatorshash tables and pseudorandom number generators
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What type of number generators make use of prime numbers?
Ground Truth Answers: pseudorandom number generatorspseudorandompseudorandompseudorandom
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Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.
Assuming p is a prime other than 2 or 5, then, according to Fermat's theorem, what type of decimal will 1/p always be?
Ground Truth Answers: a recurring decimalrecurringrecurringrecurringrecurring
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According to Fermat's theorem, what period does 1/p always have assuming p is prime that is not 2 or 5?
Ground Truth Answers: p − 1p − 1p − 1 or a divisor of p − 1p − 1 or a divisor of p − 1p − 1 or a divisor of p − 1
Prediction:
According to Wilson's theorem, what factorial must be divisible by p if some integer p > 1 is to be considered prime?
Ground Truth Answers: (p − 1)! + 1(p − 1)! + 1(p − 1)! + 1(p − 1)! + 1(p − 1)! + 1
Prediction:
According to Wilson's theorem, what factorial must be divisible by n if some integer n > 4 is to be considered composite?
Ground Truth Answers: (n − 1)!(n − 1)!(n − 1)!(n − 1)!(n − 1)!
Prediction:
What condition what must be satisfied in order for 1/p to be expressed in base q instead of base 10 and still have a period of p - 1?
Ground Truth Answers: p is not a prime factor of qp is not a prime factor of qp is not a prime factor of qp is not a prime factor of q.p is not a prime factor of q.
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Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.
What is one type of public key cryptography algorithm?
Ground Truth Answers: RSARSARSARSA
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What is another type of public key cryptography algorithm?
Ground Truth Answers: the Diffie–Hellman key exchangeDiffie–HellmanDiffie–Hellman key exchangeDiffie–Hellman key exchange
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How many bits are often in the primes used for RSA public key cryptography algorithms?
Ground Truth Answers: 512-bit512512512
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On what type of exponentiation does the Diffie–Hellman key exchange depend on?
Ground Truth Answers: modular exponentiationmodularmodularmodular
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How many bits are typically used in the primes for the Diffie–Hellman key exchange?
Ground Truth Answers: 1024-bit102410241024
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The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.
What type of insect employs the use of prime numbers in its evolutionary strategy?
Ground Truth Answers: cicadascicadascicadascicadas
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Where do cicadas spend the majority of their lives?
Ground Truth Answers: as grubs undergroundundergroundundergroundunderground
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Other than 7 and 13, what other year interval do cicadas pupate?
Ground Truth Answers: 17 years171717
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What is the logic behind the cicadas prime number evolutionary strategy?
Ground Truth Answers: make it very difficult for predators to evolve that could specialize as predatorsdifficult for predators to evolve that could specialize as predators on Magicicadasthe prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadasthe prime number intervals between emergences make it very difficult for predators to evolve
Prediction:
How much larger would cicada predator populations be if cicada outbreaks occurred at 14 and 15 year intervals?
Ground Truth Answers: up to 2% higher2%2%2%
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The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.
What does the word prime generally suggest?
Ground Truth Answers: indecomposabilityminimalityminimality or indecomposabilityminimality or indecomposability
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For a field F containing 0 and 1, what would be the prime field?
Ground Truth Answers: the smallest subfieldthe smallest subfieldQ or the finite field with p elementsthe smallest subfield
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How can any knot be distinctively indicated?
Ground Truth Answers: as a connected sum of prime knotsas a connected sum of prime knotsas a connected sum of prime knotsas a connected sum of prime knots
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What is an additional meaning intended when the word prime is used?
Ground Truth Answers: any object can be, essentially uniquely, decomposed into its prime componentsany object can be, essentially uniquely, decomposed into its prime componentsany object can be, essentially uniquely, decomposed into its prime componentsany object can be, essentially uniquely, decomposed into its prime components
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What does it mean for a knot to be considered indecomposable?
Ground Truth Answers: it cannot be written as the knot sum of two nontrivial knotscannot be written as the knot sum of two nontrivial knotscannot be written as the knot sum of two nontrivial knotsit cannot be written as the knot sum of two nontrivial knots
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Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is
What is the name of an algebraic structure in which addition, subtraction and multiplication are defined?
Ground Truth Answers: commutative ring Rcommutative ringring Rcommutative ring R
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What is one general concept that applies to elements of commutative rings?
Ground Truth Answers: prime elementsprime elementsprime elementsprime elements
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What is another general concept that applies to elements of commutative rings?
Ground Truth Answers: irreducible elementsirreducible elementsirreducible elementsirreducible elements
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What is one condition that an element p of R must satisfy in order to be considered a prime element?
Ground Truth Answers: it is neither zero nor a unitneither zero nor a unitit is neither zero nor a unitit is neither zero nor a unit
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Under what condition is an element irreducible?
Ground Truth Answers: cannot be written as a product of two ring elements that are not unitsnot a unit and cannot be written as a product of two ring elements that are not units.it is not a unit and cannot be written as a product of two ring elements that are not unitsit is not a unit and cannot be written as a product of two ring elements that are not units
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The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.
What theorem remains valid in unique factorization domains?
Ground Truth Answers: The fundamental theorem of arithmetictheorem of arithmeticfundamental theorem of arithmeticThe fundamental theorem of arithmetic
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What is one example of a unique factorization domain?
Ground Truth Answers: the Gaussian integers Z[i]Gaussian integersGaussian integers Z[i],Gaussian integers Z[i]
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What form do complex Gaussian integers have?
Ground Truth Answers: a + bia + bia + bia + bi
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What do a and b represent in a Gaussian integer expression?
Ground Truth Answers: arbitrary integersarbitrary integersarbitrary integersarbitrary integers
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Of what form are rational primes?
Ground Truth Answers: 4k + 34k + 3Z
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In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.
In what theory is the idea of a number exchanged with that of an ideal?
Ground Truth Answers: In ring theoryringring theoryring theory
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What type of ideals generalize prime elements?
Ground Truth Answers: Prime idealsPrimePrime idealsPrime ideals
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What type of number theory utilizes and studies prime ideals?
Ground Truth Answers: algebraic number theoryalgebraicalgebraicalgebraic
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Which theorem can be simplified to the Lasker–Noether theorem?
Ground Truth Answers: The fundamental theorem of arithmetictheorem of arithmeticfundamental theorem of arithmeticThe fundamental theorem of arithmetic
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What type of commutative ring does the Lasker–Noether theorem express every ideal as an intersection of primary ideals in?
Ground Truth Answers: a Noetherian commutative ringNoetherianNoetherian commutative ringNoetherian
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Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. Such ramification questions occur even in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations
What are the points of algebro-geometric objects?
Ground Truth Answers: Prime idealsPrime idealsPrime idealsPrime ideals
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What does factorization of prime ideals approximate?
Ground Truth Answers: ramification in geometryramificationramificationramification in geometry
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In what type of ring can prime ideals be used for validating quadratic reciprocity?
Ground Truth Answers: ring of integers of quadratic number fieldsintegers of quadratic number fieldsintegers of quadratic number fieldsthe ring of integers of quadratic number fields
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What does quadratic reciprocity seek to achieve?
Ground Truth Answers: the solvability of quadratic equationssolvability of quadratic equationssolvability of quadratic equationsthe solvability of quadratic equations
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In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual absolute value (also referred to as the infinite prime). While completing Q (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to the p-adic norm |−|p yields the field of p-adic numbers. These are essentially all possible ways to complete Q, by Ostrowski's theorem. Certain arithmetic questions related to Q or more general global fields may be transferred back and forth to the completed (or local) fields. This local-global principle again underlines the importance of primes to number theory.
What happens to the norm when a number is multiplied by p?
Ground Truth Answers: norm gets smallergets smallergets smallergets smaller
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To what may general global fields be transferred to or from?
Ground Truth Answers: completed (or local) fieldscompleted (or local) fieldsthe completed (or local) fieldsthe completed (or local) fields
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Completing Q with respect to what will produce the field of real numbers?
Ground Truth Answers: the absolute valuethe absolute valueabsolute valuethe absolute value
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What principle highlights the significance of primes in number theory
Ground Truth Answers: local-global principlelocal-globallocal-global principlelocal-global principle
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Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".
Which French composer wrote ametrical music using prime numbers?
Ground Truth Answers: Olivier MessiaenOlivier MessiaenOlivier MessiaenOlivier Messiaen
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What is one work by Olivier Messiaen?
Ground Truth Answers: La Nativité du SeigneurLa Nativité du SeigneurLa Nativité du SeigneurLa Nativité du Seigneur
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What is another piece created by Olivier Messiaen?
Ground Truth Answers: Quatre études de rythmeQuatre études de rythmeQuatre études de rythmeQuatre études de rythme
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In which etude of Neumes rythmiques do the primes 41, 43, 47 and 53 appear in?
Ground Truth Answers: the third étudethirdthirdthird
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Messiaen says that composition with prime numbers was inspired by what?
Ground Truth Answers: the movements of naturethe movements of nature, movements of free and unequal durationsthe movements of nature, movements of free and unequal durationsthe movements of nature, movements of free and unequal durations
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