HMMT 二月 2007 · 冲刺赛 · 第 12 题

英文解析

12. [ 8 ] Let A denote the answer to problem 11. Determine the smallest prime p such that the arithmeticsequence p, p + A , p + 2 A , . . . begins with the largest possible number of primes.11
    11 11
    Answer: 7 . First, note that the maximal number of initial primes is bounded above by the smallestprime not dividing A , with equality possible only if p is this prime. For, if q is the smallest primenot dividing A , then the first q terms of the arithmetic sequence determine a complete residue class 11
    modulo q , and the multiple of q is nonprime unless it equals q . If q < A , then q must appear first in 11
    the sequence, and thus divide the ( q + 1)st term. If q > A , then A = 2 and q = 3 by Bertrand’s 11
    11 11
    postulate, so q must appear first by inspection.
    Now since A = 30, the bound is 7. In fact, 7, 37, 67, 97, 127, and 157 are prime, but 187 is not.
    Then on the one hand, our bound of seven initial primes is not realizable. On the other hand, this 11
    implies an upper bound of six, and this bound is achieved by p = 7. Smaller primes p yield only oneinitial prime, so 7 is the answer.
    Remarks. A number of famous theorems are concerned with the distribution of prime numbers. Fortwo relatively prime positive integers a and b, the arithmetic progression a, a + b, a + 2 b, . . . containsinfinitely many primes, a result known as Dirichlet’s theorem . It was shown recently (c. 2004) that there exist arbitrarily long arithmetic progressions consisting of primes only. Bertrand’s postulate statesthat for any positive integer n , there exists a prime p such that n < p ≤ 2 n . This is an unfortunatemisnomer, as the statement is known to be true. As with many theorems concerning the distributionsof primes, these results are easily stated in elementary terms, concealing elaborate proofs.
    There is just one triple of possible ( A , A , A ) of answers to these three problems. Your team
    10 11 12
    will receive credit only for answers matching these. (So, for example, submitting a wrong answer for