Ruth-Aaron Pair
A Ruth-Aaron pair is a pair of consecutive numbers 
such that
the sums of the prime factors of 
and 
are equal. They
are so named because they were inspired by the pair (714, 715) corresponding to Hank
Aaron's record-breaking 715th home run on April 8, 1974, breaking Babe Ruth's earlier
record of 714 (Pomerance 2002; Hoffman 1998, pp. 179-181). These have the factorizations
 |  |  |
(1)
|
 |  |  |
(2)
|
and 
.
If multiplicities are not counted (so that a factor of 
counts only
a single 2), then the first few 
s giving Ruth-Aaron
pairs are 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, ... (OEIS A006145),
corresponding to the sums 5, 5, 7, 18, 15, 20, 44, 46, 29, ... (OEIS A006146).
If multiplicities are counted (so that a factor of 
counts as 
, then the first few 
s giving Ruth-Aaron
pairs are 5, 8, 15, 77, 125, 714, 948, ... (OEIS A039752),
corresponding to the sums 5, 6, 8, 18, 15, 29, 86, ... (OEIS A054378).
The numbers of such 
less than 
, 2, ... are
2, 4, 7, 20, 57, 149, 523, ... (OEIS A101805).
Nelson et al. (1974) showed that a conjecture of Schinzel on simultaneous prime values of polynomials known as "Schinzel's Hypothesis H" would imply the existence of infinitely many Ruth-Aaron pairs. This conjecture remains open (Pomerance 2002), despite an erroneous claimed proof by Erdős claimed by Hoffman (1998, pp. 180-181).
Nelson et al. (1974) also conjectured that Ruth-Aaron pairs were sparse (i.e., have density 0), a conjecture subsequently proved by Erdős and Pomerance (1978),
who showed that if a Ruth-Aaron number is defined as a number 
such that 
where 
is the sum
of prime factors of 
taken with multiplicity, then the number
of Ruth-Aaron numbers up to 
is
 |
(3)
|
which can be improved to 
. Pomerance
(2002) subsequently improved this bound to
 |
(4)
|
thus establishing the fact that the sum of the reciprocals of the Ruth-Aaron numbers is bounded. In fact,
 |
(5)
|
Babai, L.; Pomerance, C. and Vértesi, P. "The Mathematics of Paul Erdős." Notices Amer. Math. Soc. 45, 19-23, 1998.
Drost, J. L. "Ruth/Aaron Pairs." J. Recr. Math. 28, No. 2, pp. 120-122.
Erdős, P. and Pomerance, C. "On the Largest Prime Factors of 
and 
." Aeq.
Math. 17, 311-321, 1978.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.
Mackenzie, D. "Mathematics: Homage to an Itinerant Master." Science 275, 759, 1997.
Nelson, C.; Penney, D. E.; and Pomerance, C. "714 and 715." J. Recr. Math. 7, 87-89, 1974.
Peterson, I. "Ivars Peterson's MathLand: Playing with Ruth-Aaron Pairs." https://www.maa.org/mathland/mathland_6_30.html.
Peterson, I. "MathTrek: Playing with Ruth-Aaron Pairs." Sci. News 168, Aug. 6, 2005. https://www.sciencenews.org/articles/20050806/mathtrek.asp.
Pomerance, C. "Ruth-Aaron Numbers Revisited." In Paul Erdős and his Mathematics. I. Papers from the Conference Held in Budapest, July 4-11, 1999 (Ed. G. Halász, L. Lovász, M. Simonovits, and V. T. Sós). Berlin: Springer-Verlag, pp. 567-579, 2002. https://cm.bell-labs.com/cm/ms/who/carlp/PS/aaron6.ps.
Sloane, N. J. A. Sequences A006145, A006146, A039752, A054378, and A101805 in "The On-Line Encyclopedia of Integer Sequences."
Weisstein, Eric W. "Ruth-Aaron Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ruth-AaronPair.html
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