In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
are equal:
:714 = 2 × 3 × 7 × 17,
:715 = 5 × 11 × 13,
and
: 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29.
There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization.
The name was given by
Carl Pomerance
Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number ...
for
Babe Ruth
George Herman "Babe" Ruth Jr. (February 6, 1895 – August 16, 1948) was an American professional baseball player whose career in Major League Baseball (MLB) spanned 22 seasons, from 1914 through 1935. Nicknamed "the Bambino" and "the Su ...
and
Hank Aaron
Henry Louis Aaron (February 5, 1934 – January 22, 2021), nicknamed "Hammer" or "Hammerin' Hank", was an American professional baseball right fielder who played 23 seasons in Major League Baseball (MLB), from 1954 through 1976. One of the gre ...
, as Ruth's career regular-season
home run
In baseball, a home run (abbreviated HR) is scored when the ball is hit in such a way that the batter is able to circle the bases and reach home plate safely in one play without any errors being committed by the defensive team. A home run i ...
total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
at the
University of Georgia
, mottoeng = "To teach, to serve, and to inquire into the nature of things.""To serve" was later added to the motto without changing the seal; the Latin motto directly translates as "To teach and to inquire into the nature of things."
, establ ...
at the time Aaron (a member of the nearby
Atlanta Braves
The Atlanta Braves are an American professional baseball team based in the Atlanta metropolitan area. The Braves compete in Major League Baseball (MLB) as a member club of the National League (NL) East division. The Braves were founded in Bos ...
) 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.
Examples
If only distinct
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
s are counted, the first few Ruth–Aaron pairs are:
:(
5,
6), (
24,
25), (
49,
50), (
77,
78), (
104 104 may refer to:
*104 (number), a natural number
*AD 104, a year in the 2nd century AD
* 104 BC, a year in the 2nd century BC
* 104 (MBTA bus), Massachusetts Bay Transportation Authority bus route
* Hundred and Four (or Council of 104), a Carthagin ...
,
105), (
153,
154), (369, 370), (492, 493), (714, 715), (1682, 1683), (2107, 2108)
(The lesser of each pair is listed in ).
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 ).
The intersection of the two lists begins:
:(5, 6), (77, 78), (714, 715), (5405, 5406).
(The lesser of each pair is listed in ).
Any Ruth–Aaron pair of
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
s belongs to both lists with the same sum of prime factors. The intersection also contains pairs that are not square-free, for example (7129199, 7129200) = (7×11
2×19×443, 2
4×3×5
2×13×457). Here 7+11+19+443 = 2+3+5+13+457 = 480, and also 7+11+11+19+443 = 2+2+2+2+3+5+5+13+457 = 491.
Density
Ruth-Aaron pairs are sparse (that is, they have density 0). This was conjectured by Nelson et al. in 1974 and proven in 1978 by
Paul Erdős and Pomerance.
[Erdős, P. and Pomerance, C. "On the Largest Prime Factors of n and n+1." ]Aequationes Mathematicae
''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topic ...
17, 311-321, 1978.
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.
only the 4 above triplets are known.
See also
*
Maris–McGwire–Sosa pair
References
External links
*
"Ruth–Aaron Triplets"an
''The prime puzzles & problems connection''. Retrieved November 9, 2006.
{{DEFAULTSORT:Ruth-Aaron pair
Prime numbers
Babe Ruth