Hilbert-Euler problem

A generalization of the Goldbach–Euler problem (cf. Goldbach problem) according to which any even natural number larger than 2 can be represented as the sum of two prime numbers.

The Hilbert–Euler problem was formulated by D. Hilbert [1] as part of a problem on prime numbers (Hilbert's eighth problem). In fact, Hilbert advanced the hypothesis according to which the problem of the distribution of prime numbers allows one to solve both the Goldbach–Euler problem and the more general problem of solvability of the linear Diophantine equation

$$ax+by+c=0$$

in prime numbers, with given prime mutually-prime coefficients.

A special case of the Hilbert–Euler problem is the problem of twins. No solution has as yet (1989) been found to the problem, except for trivial cases. See also Additive problems.

References

[1]	"Hilbert's problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)

References

[a1]	F.E. Browder (ed.) , Mathematical developments arising from Hilbert problems , Proc. Symp. Pure Math. , 28 , Amer. Math. Soc. (1976)