Ruth-Aaron Pair Checker

Sum of prime factors match

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About Ruth-Aaron Pair Checker

A Ruth-Aaron pair checker testing if n and n+1 have equal sum of prime factors (with or without multiplicity). Named after baseball players Babe Ruth (714) and Hank Aaron (715). Shows factorizations and nearby pairs. Client-side.

Ruth-Aaron Pair Checker Features

  • Pair check
  • Factorizations
  • With/without multiplicity
  • Sequence
  • Baseball history
Ruth-Aaron pair: consecutive n, n+1 with equal sum of prime factors. Named after 714 (Babe Ruth's home run record) and 715 (Hank Aaron's record-breaking hit). 714=2·3·7·17, sum=29. 715=5·11·13, sum=29. Two versions: with/without multiplicity.

How to Use

Enter n:

  • Pair?: Same prime factor sum?
  • Factors: Both factorizations
  • Version: With/without multiplicity

Baseball Connection

On April 8, 1974, Hank Aaron hit his 715th home run, breaking Babe Ruth's record of 714. Carl Pomerance noticed 714 and 715 share the same prime factor sum (29), coining the term 'Ruth-Aaron pair'.

How Common?

Erdős and Pomerance conjectured infinitely many Ruth-Aaron pairs exist. First few: (5,6), (24,25), (49,50), (77,78), (104,105), (714,715)... About 1 in 100 pairs up to 10,000.

Step-by-Step Instructions

  1. 1Enter number.
  2. 2Check pair.
  3. 3Compare factorizations.
  4. 4View sum.
  5. 5Browse pairs.

Ruth-Aaron Pair Checker — Frequently Asked Questions

What's the difference between the two versions?+

Without multiplicity: sum distinct primes (e.g., 8=2, sum=2). With multiplicity: sum with exponents (8=2³, sum=2+2+2=6). Some pairs work for one version but not the other. (714,715) works for both!

Are there infinitely many Ruth-Aaron pairs?+

Conjectured yes by Erdős and Pomerance, but unproven! They showed the count up to N is at most N/(log N)^ε for some ε, suggesting they thin out. But no one has proven they're infinite or finite.

What are some large Ruth-Aaron pairs?+

They become rarer as numbers grow. The pair (18490,18491) is an example: 18490=2·5·43·43, sum=93; 18491=13·1423, sum... Checking requires factorization. Nelson, Penney, and Pomerance computed many pairs.

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