HMMT 十一月 2015 · 冲刺赛 · 第 36 题
HMMT November 2015 — Guts Round — Problem 36
题目详情
- [ 20 ] Consider the following seven false conjectures with absurdly high counterexamples. Pick any subset of them, and list their labels in order of their smallest counterexample (the smallest n for which the conjecture is false) from smallest to largest. For example, if you believe that the below list is already 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 an odd number of prime factors. • E. ( Euler’s conjecture ) There is no perfect cube n that can be written as the sum of three positive cubes. • C. ( Cyclotomic ) The polynomial with minimal degree whose roots are the primitive n th roots of 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 there are 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 sum of 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 n k 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.
解析
- [ 20 ] Consider the following seven false conjectures with absurdly high counterexamples. Pick any subset of them, and list their labels in order of their smallest counterexample (the smallest n for which the conjecture is false) from smallest to largest. For example, if you believe that the below list is already 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 an odd number of prime factors. • E. ( Euler’s conjecture ) There is no perfect cube n that can be written as the sum of three positive cubes. • C. ( Cyclotomic ) The polynomial with minimal degree whose roots are the primitive n th roots of 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 there are 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 sum of 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 n k 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