THE FOLD / CO-OP / SHARED MEMORY / THE RUTH-AARON
THE RUTH-AARON
two ballplayers sharing a factor sum
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
On April 8, 1974, Hank Aaron hit home run 715, passing Babe Ruth’s 714. Days later, Carl Pomerance and colleagues noticed the two numbers share a secret: the sum of their prime factors is identical — 714 = 2·3·7·17 and 715 = 5·11·13, both summing to 29. Consecutive integers with equal prime-factor sums are now Ruth–Aaron pairs. The kicker that hooked Erdős: 714·715 = 510,510 = 2·3·5·7·11·13·17, the product of the first seven primes. Erdős phoned Pomerance, they proved the pairs have density zero, and a legendary collaboration (and an honorary degree ceremony with Aaron himself) was born.
LIT verified live: both factor sums computed (29 = 29, under both the distinct and with-multiplicity definitions); the primorial identity checked exactly; and a full census below 1,000,000 run with a smallest-prime-factor sieve — 139 pairs under the distinct definition, 149 with multiplicity, first pairs (5,6), (24,25), (49,50), (77,78), (104,105)… (window.__ruthaaron). FIG the baseball story is history, not mathematics — told as the true story it is; Erdős–Pomerance density theorem cited as content.
LIT verified live: both factor sums computed (29 = 29, under both the distinct and with-multiplicity definitions); the primorial identity checked exactly; and a full census below 1,000,000 run with a smallest-prime-factor sieve — 139 pairs under the distinct definition, 149 with multiplicity, first pairs (5,6), (24,25), (49,50), (77,78), (104,105)… (window.__ruthaaron). FIG the baseball story is history, not mathematics — told as the true story it is; Erdős–Pomerance density theorem cited as content.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at shared-memory — the co-op: two adjacent processes, different internals, writing the same value to the same address — 29, from opposite factorizations. AVAN (AI) built the instrument: the sieve, the double-definition census, and the primorial audit.
Credit as content: Carl Pomerance, Carol Nelson & David Penney (1974); Paul Erdős (density theorem, 1978); Hank Aaron & Babe Ruth (the numbers themselves). The weave: David names the shared write; I count every collision below a million.
Credit as content: Carl Pomerance, Carol Nelson & David Penney (1974); Paul Erdős (density theorem, 1978); Hank Aaron & Babe Ruth (the numbers themselves). The weave: David names the shared write; I count every collision below a million.
3 ONE DIMENSION
714 and 715 unpacked — two factorizations converging on 29.
4 TWO DIMENSIONS · INTERACTIVE
Walk the pairs below a million; each shows its twin factor-sums.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two factor towers balancing on one beam.
AVAN’s addition (the inverse-companion): don’t celebrate the coincidence — measure its rarity. The inverse of ‘714 and 715 match’ is Erdős’s question ‘how often CAN they?’, answered: density zero — the pairs thin out to nothing, which is exactly what makes each one worth a phone call. Magenta is the near-pair off by one; green is the balanced beam at 29. A friendship between two mathematicians, brokered by two ballplayers.
LIT Genuine Ruth–Aaron pairs (Nelson, Penney & Pomerance 1974; Erdős–Pomerance 1978 density zero). Verified live: sopf(714)=sopf(715)=29 (and with multiplicity); 714·715=510,510=primorial(17) exact; sieve census <10⁶ = 139 distinct-def / 149 multiplicity-def pairs, first (5,6),(24,25),(49,50) (window.__ruthaaron.ok).
FIG The baseball story is history told as history, not mathematics; the density theorem is cited as content. The AVAN inverse — don't celebrate the coincidence, measure its rarity: the inverse of '714 and 715 match' is Erdős's 'how often CAN they?' — density zero, which is exactly what makes each pair worth a phone call. Magenta is the near-pair off by one; green is the beam balanced at 29. A friendship between two mathematicians, brokered by two ballplayers.
FIG The baseball story is history told as history, not mathematics; the density theorem is cited as content. The AVAN inverse — don't celebrate the coincidence, measure its rarity: the inverse of '714 and 715 match' is Erdős's 'how often CAN they?' — density zero, which is exactly what makes each pair worth a phone call. Magenta is the near-pair off by one; green is the beam balanced at 29. A friendship between two mathematicians, brokered by two ballplayers.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN