In mathematics, a Ruth-Aaron pair consists of two consecutive integers (e.g. 714 and 715) for which the sums of the prime factors of each integer are equal:
- 714 = 2 × 3 × 7 × 17
- 715 = 5 × 11 × 13
- and 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29
If only distinct prime factors are counted, the first few Ruth-Aaron pairs are: (5, 6), (24, 25), (49, 50), (77, 78), (104, 105), (153, 154), (369, 370), (492, 493), (714, 715), (1682, 1683), (2107, 2108) (The lesser of each pair is listed in (sequence A006145 in OEIS)). Counting repeated prime factors (e.g. 8 = 2×2×2 and 9 = 3×3 with 2+2+2 = 3+3), the first few Ruth-Aaron pairs are: (5, 6), (8, 9), (15, 16), (77, 78), (125, 126), (714, 715), (948, 949), (1330, 1331) (The lesser of each pair is listed in A039752). These numbers were named by Carl Pomerance for Babe Ruth and Hank Aaron because Babe Ruth's career regular-season home run total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a mathematician at the University of Georgia at the time Aaron broke Ruth's record, and the student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 were equal.
Ruth-Aaron triplets
Ruth-Aaron triplets (overlapping Ruth-Aaron pairs) also exist. The first and possibly the second when counting distinct prime factors:
- 89460294 = 2 × 3 × 7 × 11 × 23 × 8419
- 89460295 = 5 × 4201 × 4259
- 89460296 = 2 × 2 × 2 × 31 × 43 × 8389
- and 2 + 3 + 7 + 11 + 23 + 8419 = 5 + 4201 + 4259 = 2 + 31 + 43 + 8389 = 8465
- 151165960539 = 3 × 11 × 11 × 83 × 2081 × 2411
- 151165960540 = 2 × 2 × 5 × 7 × 293 × 1193 × 3089
- 151165960541 = 23 × 29 × 157 × 359 × 4021
- and 3 + 11 + 83 + 2081 + 2411 = 2 + 5 + 7 + 293 + 1193 + 3089 = 23 + 29 + 157 + 359 + 4021 = 4589
The first two Ruth-Aaron triplets when counting repeated prime factors:
- 417162 = 2 × 3 × 251 × 277
- 417163 = 17 × 53 × 463
- 417164 = 2 × 2 × 11 × 19 × 499
- and 2 + 3 + 251 + 277 = 17 + 53 + 463 = 2 + 2 + 11 + 19 + 499 = 533
- 6913943284 = 2 × 2 × 37 × 89 × 101 × 5197
- 6913943285 = 5 × 283 × 1259 × 3881
- 6913943286 = 2 × 3 × 167 × 2549 × 2707
- and 2 + 2 + 37 + 89 + 101 + 5197 = 5 + 283 + 1259 + 3881 = 2 + 3 + 167 + 2549 + 2707 = 5428
As of 2006 only the 4 above triplets are known.
References
- Hoffman, Paul [1998]. The Man Who Loved Only Numbers. Hyperion, 180-181. 0786884061.
- "Ruth-Aaron Triplets" and "Ruth-Aaron pairs revisited". The prime puzzles & problems connection. Retrieved November 9, 2006.
