HMMT 十一月 2015 · 冲刺赛 · 第 36 题

英文解析

36. [ 20 ] Consider the following seven false conjectures with absurdly high counterexamples. Pick anysubset of them, and list their labels in order of their smallest counterexample (the smallest n for whichthe conjecture is false) from smallest to largest. For example, if you believe that the below list isalready ordered by counterexample size, you should write ”PECRSGA”.
    • P. ( Polya’s conjecture ) For any integer n , at least half of the natural numbers below n have anodd number of prime factors.
    • E. ( Euler’s conjecture ) There is no perfect cube n that can be written as the sum of threepositive cubes.
    • C. ( Cyclotomic ) The polynomial with minimal degree whose roots are the primitive n th rootsof unity has all coefficients equal to -1, 0, or 1.
    • R. ( Prime race ) For any integer n , there are more primes below n equal to 2 (mod 3) than thereare equal to 1 (mod 3).
    17 17
    • S. ( Seventeen conjecture ) For any integer n , n + 9 and ( n + 1) + 9 are relatively prime.
    • G. ( Goldbach’s (other) conjecture ) Any odd composite integer n can be written as the sumof a prime and twice a square.
    2 2 2
    1+ a + a + ... + a
    1 2 k
    • A. ( Average square ) Let a = 1 and a = . Then a is an integer for any n .
    1 k +1 nk
    If your answer is a list of 4 ≤ n ≤ 7 labels in the correct order, your score will be ( n − 2)( n − 3).
    Otherwise, it will be 0.
    Proposed by: Alexander Katz
    Answer: ACGPRES
    The smallest counterexamples are:
    • Polya’s conjecture: 906,150,257
    • Euler’s sum of powers: 31,858,749,840,007,945,920,321
    • Cyclotomic polynomials: 105
    • Prime race: 23,338,590,792
    • Seventeen conjecture: 8,424,432,925,592,889,329,288,197,322,308,900,672,459,420,460,792,433
    • Goldbach’s other conjecture: 5777
    • Average square: 44