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Featureist arbitrarily long arithmetic progressions consisting only ofprimes. As one would expect, the method leads to new questions– one of them is related to the case when k polynomialsp i (x) ∈ Z[X], i = 1, 2,...,k, are given and one wishes to knowif there are infinitely many integers x, m such that x + p i (m)are all simultaneously prime. The result was established in2006 by Tao and Ziegler and it obviously implies the one onarithmetic progressions.The problem is more intricate if one wants to count thefrequency of arithmetic sequences or, more generally, of givenpatterns of primes. In this context, an accurate analysis basedon the Cramér model leads to the famous conjecture of Hardyand Littlewood: for integer sets H = {h 1 , ..., h k } they first defineda constant Σ(H) ∈ R ≥0 , which vanishes if there is no(k + 1)-tuple of primes with mutual distances h i ; otherwise itis a constant which is deduced from the Poisson distributionof gaps (see [9] for more details), such that the following isconsistent with the Cramér model:Conjecture 3. Let H = {h 1 , ..., h k } be a set of positive integerssuch that Σ(H) 0. Thenx|{n ≤ x : n + h 1 , ..., n + h k ∈ P}| ∼ Σ(H)(log x) . kThis is what one would expect and it suggests not onlythat there should be infinitely many Sophie-Germain pairsbut that their occurence is quite frequent, about one in everylog(x) 2 , on average. A stronger version of this conjecture isdue to Bateman and Horn and it replaces the linear polynomialsf i (x) = x + h i , for which the Hardy Littlewood conjecturerequires that f i (n) ∈ P for all i, by some predetermined polynomialsf i (x) of arbitrary degree. The claim of the conjectureis then similar, with the same distribution function dependingonly on k and a structural constant that replaces Σ(H).Particular problems and conjecturesA variety of problems related to prime gaps or to distributionsof primes have been proposed, sometimes in the hopethat these might represent a simpler, more accessible problem.The hope did not often come true.The Hypothesis H of Schinzel [10] states:Conjecture 4. Consider s polynomials f i (x) ∈ Z[X]; i =1, 2,...,s with positive leading coefficients and such that theproductsF(X) = f i (x)is not divisible, as a polynomial, by any integer different from±1. Then there is at least one integer x for which all the polynomialsf i (x) take prime values.Apparently, this is only a more accessible version of theBateman-Horn conjecture.Dorin Andrică [2] proposed during his years of study thefollowing conjecture, which was sufficiently supported by numericalevidence:Conjecture 5. Let p n ∈ P denote as usual the n−th prime.Then, for all n ∈ N, the distance√pn+1 − √ p n < 1.i=1Equivalently, the gapg n = p n+1 − p n < 2 √ p n + 1.This appears as if it was an easy exercise, given that we“know" from the Cramér model that the typical gap has lengthlog(p n ) and we even expect that no larger gaps than log 2 (p n )exist, with little to say about exponential gaps like O( √ p n ).But not only does the estimate lay in the order of magnitudeof what one can prove by assuming the Riemann Hypothesisbut it appears that proving the one is essentially equivalent toproving the other.An apparently even simpler conjecture was formulated byMichael Th. Rassias [8], at the age of 14, while preparingfor the International Mathematical Olympiad that was heldin Tokyo in 2003. Rassias’ numerically supported conjectureclaims:Conjecture 6. For any prime p > 2 there are two otherprimes p 1 < p 2 such thatp = p 1 + p 2 + 1.p 1We have here a surprising feature of presenting a prime asa quotient; however, after some algebraic manipulation, werecover the expression(p − 1)p 1 = p 2 + 1,thus obtaining an interesting combination of a generalised SophieGermain twin problemp 2 = 2ap 1 − 1,strengthened by the additional condition that 2a+1 be a primenumber too. We have seen that such questions are caught bythe Hardy–Littlewood conjecture. One may ask if Rassias’conjecture is to some extent simpler than the general Hardy–Littlewood conjecture or its special case concerning distributionof generalised Sophie-Germain pairs p, 2ap+1 ∈ P? Thisis not likely to be true.A further related problem, much appreciated by amateurnumber theorists in search of interesting computations, is relatedto Cunningham chains, i.e. sequences of primesp i+1 = mp i + n, i = 1, 2,...,k − 1,for fixed coprime m, n ∈ N >1 .As for the large arithmetic progressions of primes, there arecomputing competitions for the longest Cunningham chain orfor the one built up of the largest primes – but unlike thebreakthrough of Green and Tao, there is no general resultknown on large Cunningham chains to date. Rassias’ conjecturecan also be stated in terms of Cunningham chains,namely: there exist Cunningham chains with parameters 2a, −1for a such that 2a − 1 = p is prime.References[1] http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records.[2] T. Andreescu and D. Andrica: Number Theory, Birkhäuser,Boston, Basel, Berlin (2009), p. 12.[3] H. Dubner: Large Sophie-Germain primes, Math. Comp.65(1996), pp. 393–396.EMS Newsletter June 2014 15