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Fun with Elementary Number Theory! The Ruth-Aaron Pair

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You know Henry Aaron, who was born on this very date in 1934, for his legendary assault on the most cherished record in baseball. You know Carl Pomerance, who was born in 1944, for his legendary quadratic sieve algorithm and his nearly-as-legendary Adleman–Pomerance–Rumely primality test. But did you know that the lives of these two legends are forever intertwined? Consider:

Let S(n) denote the sum of the prime factors of n taken with multiplicity. We say that n is a Ruth-Aaron number if S(n) = S(n+1) in honor of the famous American baseball players Babe Ruth and Hank Aaron. Ruth’s lifetime homerun record was 714, which Aaron broke on April 8, 1974, by hitting number 715 toward his own eventual record of 755. Note that S(714) = S(715). Erdos and I proved in 1978, in our first joint paper, that the number of Ruth-Aaron numbers up to x is O(x log log x log log log x / log x)…

You will obviously want to read this groundbreaking paper in its entirety, but for those of you who are pressed for time, I’ll go ahead and cut right to the chase:

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Juicy stuff! I trust Mr. Aaron would appreciate honoring his birthday in such a fashion; after all, he is no stranger to number theory, famously boasting an Erdos-Bacon number of 3. Happy birthday, Hank! Hope it’s a prime one.

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Aaron explains the Chinese remainder theorem to Erdos, by means of a Chinese fingertrap.