Bunjakowski conjecture

The Bunjakowski conjecture is an open conjecture of number theory .

There is no integer polynomial in a variable that only generates prime numbers when inserting natural numbers ( Adrien-Marie Legendre ). But one can ask oneself whether there are those that have an infinite number of prime numbers as values.

In 1857 Viktor Jakowlewitsch Bunjakowski specified three necessary conditions that a polynomial of positive degree and with integer coefficients must have so that among the values ( ) there are infinitely many prime numbers: ${\displaystyle f(x)}$ ${\displaystyle f(n)}$ ${\displaystyle n\in \mathbb {N} }$

1. the leading coefficient is positive.
2. the polynomial is irreducible over the whole numbers. ${\displaystyle f(x)}$
3. the ( ) are relatively prime, that is, their greatest common factor (gcd) is 1. ${\displaystyle f(n)}$ ${\displaystyle n\in \mathbb {N} }$

Bunjakowski hypothesized that the conditions are also sufficient, that is, every polynomial that satisfies the three conditions has an infinite number of prime numbers as values.

The coefficients of the polynomial must be relatively prime (gcd equal to 1) for the equation to have more than two nontrivial prime number solutions. This also follows from condition 3. An example of a polynomial that satisfies conditions 1 and 2, but not 3, is that has only even values. The only prime value is for . If the coefficients of the polynomial are relatively prime (gcd equals 1), this does not mean, conversely, that condition 3 applies (as the same example shows). ${\displaystyle f(x)=x^{2}+x+2=x(x+1)+2}$ ${\displaystyle 2}$ ${\displaystyle x=0}$

To find out whether condition 3 is true, one only needs to find such that and are relatively prime. Then they can not have a GCF , otherwise it would also be a common factor for and . ${\displaystyle m\neq n}$ ${\displaystyle f(m)}$ ${\displaystyle f(n)}$ ${\displaystyle f(n)}$ ${\displaystyle b\neq 1}$ ${\displaystyle f(n)}$ ${\displaystyle f(m)}$

For the other individual conditions:

1. If the leading coefficient were negative, the number would be sufficiently large and therefore not a prime number. You can omit the condition if you also allow negative prime numbers as values. ${\displaystyle f(n)<0}$ ${\displaystyle n}$
2. If for all (reducible), whereby integer polynomials are not identical , then it would be composite for sufficiently large . Because there are only a finite number of solutions for , and accordingly for . ${\displaystyle f(n)=g(n)\cdot h(n)}$ ${\displaystyle n}$ ${\displaystyle f(n),g(n)}$ ${\displaystyle \pm 1}$ ${\displaystyle f(n)}$ ${\displaystyle n}$ ${\displaystyle g(n)=0}$ ${\displaystyle g(n)=\pm 1}$ ${\displaystyle h}$

An example of polynomials over the integers that meet Bunjakowski's three conditions are the circle division polynomials . Another example is (that this yields an infinite number of prime numbers, assumed Leonhard Euler and is one of the Landau problems and also follows from the fifth Hardy-Littlewood conjecture). ${\displaystyle x^{2}+1}$

Bunjakowski's assumption is open. It is only proven in the case of polynomials of the first degree ( Dirichlet's prime number theorem ). But there is numerical support for the guess in the other cases. ${\displaystyle f(x)=ax+b}$

Polynomials with the three properties listed above and degrees greater than 1 are also called Bunjakowski polynomials. It is not known whether all Bunjakowski polynomials have at least one prime solution.

Various conclusions from the Bunjakowski conjecture and its generalization to systems of several irreducible polynomials by Andrzej Schinzel and Wacław Sierpiński are presented in a book by Paulo Ribenboim .

Web links

• Bounikowsky conjecture , Mathworld (by Ed Pegg Jr.)

Individual evidence

1. ^ Bunjakowski, Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la de composition des entiers en facteurs, Sc. Math. Phys. 6, 1857, pp. 305-329
2. ↑ Sierpinski, Schinzel, Sur certaines hypothèses concernant les nombres premieres, Acta Arithmetica, Volume 4, 1958, pp. 185-208, Volume 5, 1959, p. 259, comment on this, Volume 7, 1961, pp. 1-8
3. ↑ Ribenboim, The World of Prime Numbers, Springer, 2nd edition, 2011