Ruth-Aaron pair

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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.

[edit] 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.

[edit] References

  • Hoffman, Paul [1998]. The Man Who Loved Only Numbers. Hyperion, 180-181. 0786884061. 
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