Download On April 8, 1974, Hank Aaron hit his 715th (of 755) home run thus

On April 8, 1974, Hank Aaron hit his 715th (of 755) home run thus breaking Babe Ruth’s longstanding record. At that time Carl Pomerance of the University of Georgia noticed that the product of
714 and 715 is also the product of the first seven primes.
714715  2 x 3 x 5 x 7 x 11 x 13 x 17
He challenged a colleague, David Penny, to look for interesting properties of these two numbers. Both
Pomerance and Penny made the above observation. But Penny’s student, Jeremy Jordon, observed the
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sum of the prime factors of 714 and 715 were also equal.
Factors of 714 :
2,3,7,17
Factors of 715 :
5,11,13
2 + 3 + 7 + 17 = 29
5 + 11 + 13 = 29
Note that in factoring 714 and 715, the primes are each unique and only of multiplicity 1. This led to
an investigation of factoring consecutive integers that had multiple factors of the same prime, but
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added to the same sum. Results emerged quickly.
Factors of 8 :
2,2,2
Factors of 9 :
3,3
2 +2 +2 = 6
3 +3 = 6
Pomerance, Penny and Carol Nelson then published an article stating that if one accepted an earlier
Schnizel’s Hypothesis H (1958), then there would be infinitely many pairs of integers n and n + 1
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for which the sums are equal.
S(n) = S( n + 1)
Today we call these “Ruth-Aaron” numbers.
Please note the name “Ruth-Aaron number” applies to a pair of integers both with and without singular
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prime factors.
In particular, if 2n  1, 8n  5, 48n 2  24n 1, and 48n 2  30n 1 are all prime, then the product
of the middle two is a “Ruth-Aaron” number. Moreover, there are relatively few, or technically
speaking, the density of Ruth-Aaron numbers is 0. This caught the attention of the famous number
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theorist Paul Erdös. He wrote Pomerance telling him he could prove it - - -leading to the first of over
40 joint papers between the two.
References:

P. Erdös and C. Pomerance, On the largest prime factors of n and n + 1,
Aequations Mathematicae 17, (1978), 311-321.

C. Nelson, D. E. Penny and C.Pomerance, 714 and 715, J. Recreational Math.,
7(1974), 87-89.

C. Pomerance, Ruth-Aaron Numbers Revisited, Paul Erdös and His Mathematic I,
Budapest 2002, pp. 567-579, Mathematical Studies, 11, Bolyai Society.
Schinzel, A. and Sierpinski, W., Sur certaines hypothèses concernant les nombres premiers.
Remarque. Acta Arithm. 4, 185-208, 1958.