Bunyakovsky conjecture

Bunyakovsky conjecture
The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial in one variable with integer coefficients to give infinitely many prime values in the sequence It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for to have the desired prime-producing property:
the leading coefficient is positive,
the polynomial is irreducible over the rationals (and integers), and
the values have no common factor. (In particular, the coefficients of should be relatively prime.)
Bunyakovsky's conjecture is that these conditions are sufficient: if satisfies (1)–(3), then is prime for infinitely many positive integers .
A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial that satisfies (1)–(3), is prime for at least one positive integer : but then, since the translated polynomial still satisfies (1)–(3), in view of the weaker statement is prime for at least one positive integer , so that is indeed prime for infinitely many positive integers . Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory.
Discussion of three conditions
The first condition is necessary because if the leading coefficient is negative then for all large , and thus is not a (positive) prime number for large positive integers . (This merely satisfies the sign convention that primes are positive.)
The second condition is necessary because if where the polynomials and have integer coefficients, then we have for all integers ; but and take the values 0 and only finitely many times, so is composite for all large .
The second condition also fails for the polynomials reducible over the rationals.
For example, the integer-valued polynomial doesn't satisfy the condition (2) since , so at least one of the latter two factors must be a divisor of in order to have prime, which holds only if . The corresponding values are , so these are the only such primes for integral since all of these numbers are prime. This isn't a counterexample to Bunyakovsky conjecture since the condition (2) fails.
The third condition, that the numbers have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample. Consider , which has positive leading coefficient and is irreducible, and the coefficients are relatively prime; however is even for all integers , and so is prime only finitely many times (namely at , when ).
In practice, the easiest way to verify the third condition is to find one pair of positive integers and such that and are relatively prime. In general, for any integer-valued polynomial we can use for any integer , so the gcd is given by values of at any consecutive integers. In the example above, we have and so the gcd is , which implies that has even values on the integers.
Alternatively, when an integer polynomial is written in the basis of binomial coefficient polynomials
each coefficient is an integer and In the example above, this is
and the coefficients in the right side of the equation have gcd 2.
Using this gcd formula, it can be proved if and only if there are positive integers and such that and are relatively prime.
Examples
A simple quadratic polynomial
Some prime values of the polynomial are listed in the following table. (Values of form OEIS sequence ; those of form .)
That should be prime infinitely often is a problem first raised by Euler, and it is also the fifth Hardy–Littlewood conjecture and the fourth of Landau's problems. Despite the extensive numerical evidence
it is not known that this sequence extends indefinitely.
Cyclotomic polynomials
The cyclotomic polynomials for satisfy the three conditions of Bunyakovsky's conjecture, so for all k, there should be infinitely many natural numbers n such that is prime. It can be shown that if for all k, there exists an integer n > 1 with prime, then for all k, there are infinitely many natural numbers n with prime.
The following sequence gives the smallest natural number n > 1 such that is prime, for :
3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2, ... .
This sequence is known to contain some large terms: the 545th term is 2706, the 601st is 2061, and the 943rd is 2042. This case of Bunyakovsky's conjecture is widely believed, but again it is not known that the sequence extends indefinitely.
Usually, there is an integer between 2 and (where is Euler's totient function, so is the degree of ) such that is prime, but there are exceptions; the first few are
1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, ....
Partial results: only Dirichlet's theorem
To date, the only case of Bunyakovsky's conjecture that has been proved is that of polynomials of degree 1. This is Dirichlet's theorem, which states that when and are relatively prime integers there are infinitely many prime numbers . This is Bunyakovsky's conjecture for (or if ).
The third condition in Bunyakovsky's conjecture for a linear polynomial is equivalent to and being relatively prime.
No single case of Bunyakovsky's conjecture for degree greater than 1 is proved, although numerical evidence in higher degree is consistent with the conjecture.
Generalized Bunyakovsky conjecture
Given polynomials with positive degrees and integer coefficients, each satisfying the three conditions, assume that for any prime there is an such that none of the values of the polynomials at are divisible by . Given these assumptions, it is conjectured that there are infinitely many positive integers such that all values of these polynomials at are prime.
Note that the polynomials do not satisfy the assumption, since one of their values must be divisible by 3 for any integer . Neither do , since one of the values must be divisible by 3 for any .
On the other hand, do satisfy the assumption, and the conjecture implies the polynomials have simultaneous prime values for infinitely many positive integers .
This conjecture includes as special cases the twin prime conjecture (when , and the two polynomials are and ) as well as the infinitude of prime quadruplets (when , and the four polynomials are , and ), sexy primes (when , and the two polynomials are and ), Sophie Germain primes (when , and the two polynomials are and ), and Polignac's conjecture (when , and the two polynomials are and , with any even number). When all the polynomials have degree 1 this is Dickson's conjecture.
In fact, this conjecture is equivalent to the Generalized Dickson conjecture.
Except for Dirichlet's theorem, no case of the conjecture has been proved, including the above cases.
See also
Integer-valued polynomial
Cohn's irreducibility criterion
Schinzel's hypothesis H
Bateman–Horn conjecture
Hardy and Littlewood's conjecture F
References
Bibliography
Conjectures about prime numbers
Unsolved problems in number theory