Question 126
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 $2(\bmod 3)$ than there are equal to $1(\bmod 3)$.
- S. (Seventeen conjecture) For any integer $n$, $n^{17}+9$ and $(n+1)^{17}+9$ 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.
- A. (Average square) Let $a_{1}=1$ and $a_{k+1}=\frac{1+a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2}}{k}$. Then $a_{n}$ is an integer for any $n$.
If your answer is a list of $4 \leq n \leq 7$ labels in the correct order, your score will be $(n-2)(n-3)$. Otherwise, it will be 0.
- 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 $2(\bmod 3)$ than there are equal to $1(\bmod 3)$.
- S. (Seventeen conjecture) For any integer $n$, $n^{17}+9$ and $(n+1)^{17}+9$ 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.
- A. (Average square) Let $a_{1}=1$ and $a_{k+1}=\frac{1+a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2}}{k}$. Then $a_{n}$ is an integer for any $n$.
If your answer is a list of $4 \leq n \leq 7$ labels in the correct order, your score will be $(n-2)(n-3)$. Otherwise, it will be 0.