Sur certaines hypothèses concernant les nombres premiers

Sieving and expanders⁎⁎The first author was supported in part by NSF grant DMS-0322370. The second author was supported in part by NSF grant DMS-0111298 and DMS-0501245. The third author was supported in part by Oscar Veblen Fund (IAS) and the NSF.

“…For example, if Λ is generated by 7 6 8 7 and b = 1 1 , then there are no local obstructions, but V is contained in {(x, y): 4x 2 − 3y 2 = 1}, from which it is clear that y cannot be prime and hence V ∩ P 2 is empty. The formulation of the higher dimensional versions of Conjecture 2, as well as the generalization to this non-Abelian setting of Schinzel's hypothesis H [20] is more involved and we leave it to the long version of this paper [3]. Our aim here is to outline the key ingredients needed to develop a combinatorial sieve in this non-Abelian context and to apply it to establish versions of these conjectures with primes replaced by almost primes.…”

Families of non-congruent numbers with arbitrarily many prime factors

“…We also have that q k = p i for some i satisfying 2 i k − 1. From (8) Going further, by making use of Schinzel's hypothesis H [11], we can offer the following evidence that congruent numbers whose prime factors are of the form 8k + 3 and satisfy (2) exist whenever m is odd and m 3. A similar approach appears in the previously mentioned paper of Ono [9,Theorem 2], where the statement that a family of elliptic curves has positive rank is related to Schinzel's hypothesis H (called Bouniakowsky's conjecture).…”

“…We recall Schinzel's hypothesis H [11] which states that if a finite product Q (x) = m i=1 f i (x) of polynomials f i (x) ∈ Z[x] has no fixed divisors, then all of the f i (x) will be simultaneously prime, for infinitely many integral values of x. From this hypothesis we deduce that the two forms Thus, when m is odd and m 3, we cannot generate families of non-congruent numbers.…”

Reducibility of polynomials a 0 ( x ) + a 1 ( x ) y + a 2 ( x ) y 2 modulo p

“…[2] or [5, p. 307]) which says: If gt P Zt is irreducible over Q and N gcdfg X P Zg then there are infinitely many P N such that 1 N jgj is a prime number. The conjecture is a special case of Schinzels hypothesis H [7]. E x a m p l e 1 .…”