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Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x 2 − x + 41, are believed to produce a high density of prime numbers.
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as x 2 + 1 (known from Landau's fourth problem) attain infinitely many prime values is an important open problem. Dickson's conjecture generalizes Dirichlet's theorem to more than one polynomial.
If and are relatively prime positive integers then every factor of + is a sum of two squares. (This is the step that uses step (3.) to produce an 'infinite descent' and was Euler's Proposition 4. The proof sketched below also includes the proof of his Proposition 3).
For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10 1000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10 2000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N). [3]
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.
As far as is known, this is not possible using classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer.
A corollary of the Mason–Stothers theorem is the analog of Fermat's Last Theorem for function fields: if a(t) n + b(t) n = c(t) n for a, b, c relatively prime polynomials over a field of characteristic not dividing n and n > 2 then either at least one of a, b, or c is 0 or they are all constant.