The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
in one variable with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
to give
infinitely many prime values in the sequence
It was stated in 1857 by the
Russian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
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).
# 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
We need the first condition 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.)
We need the second condition 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 when
, in fact only at
).
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,
can be written in the basis of binomial coefficient polynomials:
:
where each
is an integer, and
For the above example, we have:
:
and the coefficients in the second formula 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
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
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
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau' ...
. Despite the extensive numerical evidence
it is not known that this sequence extends indefinitely.
Cyclotomic polynomials
The
cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primit ...
s
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 integer 2≤n≤
φ(k) such that
is prime (note that the
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
of
is φ(k)), but there are exceptions, the exception numbers k 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 number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms , , ..., with , there are infinitely many positive integers for which they are all prime, unless there is a congruen ...
.
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
*
*
*
{{Prime number conjectures
Conjectures about prime numbers
Unsolved problems in number theory