Primes with preassigned digits II

Glyn Harman, Imre Kátai
2008 Acta Arithmetica  
Dedicated to Professor W. M. Schmidt on his 75th birthday 1. Introduction. The purpose of this paper is to show that significant improvements can be made on the results given in [5] using the ideas contained in the second author's paper [8] . Indeed, the results stated in [8] anticipated Wolke's conjecture in [14] by some twenty years. The first named author was unaware of this work until it was pointed out to him by Cécile Dartyge. There are some oversights in the proofs of the results in [8]
more » ... the results in [8] which mean that the full strength of the theorems in that paper was not established at the time. However, even after correcting those proofs, what remains is strong enough to improve the work in [5] . In this paper we shall give improved versions of those proofs which establish substantially stronger results. We recall the history of this problem. In 1951 Sierpiński [11] began the investigation of prime numbers with preassigned digits. In [12] he showed that in any given base g and with given a, b with 1 ≤ a ≤ g −1, gcd(b, g) = 1, 1 ≤ b ≤ g − 1, one can find infinitely many primes p having a as its first digit and b as its last. As was shown in [5], these results are elementary deductions from deeper results on the distribution of primes. Wolke [14] considered the more difficult case where one preassigns at most two digits anywhere in the expansion of an integer with k digits, and obtained an asymptotic formula (valid for k → ∞) in this case. We will use here the notation of [8] to state the problem formally rather than what was used in [14, 5] . We suppose that an integer n is written in the standard form to base q: n = ∞ j=0 a j (n)q j , 0 ≤ a j (n) ≤ q − 1. 2000 Mathematics Subject Classification: Primary 11N05.
doi:10.4064/aa133-2-5 fatcat:l75matyhoreqzocj556iweuib4