Ruth-Aaron pair
Encyclopedia
In mathematics
, a Ruth–Aaron pair consists of two consecutive integers (e.g. 714 and 715) for which the sums of the prime factor
s of each integer
are equal:
and
If only distinct prime
factor
s 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 ).
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:
, (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:
, (77, 78), (714, 715), (5405, 5406)
(The lesser of each pair is listed in ).
Any Ruth-Aaron pair of square-free integer
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) = (7×112×19×443, 24×3×52×13×457). 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.
The name was given by Carl Pomerance
for Babe Ruth
and Hank Aaron, as 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 (a member of the nearby Atlanta Braves
) 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.
The first two Ruth–Aaron triplets when counting repeated prime factors:
only the 4 above triplets are known.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Ruth–Aaron pair consists of two consecutive integers (e.g. 714 and 715) for which the sums of the prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
s of each integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
are equal:
- 714 = 2 × 3 × 7 × 17
- 715 = 5 × 11 × 13
and
- 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29
If only distinct prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
factor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s are counted, the first few Ruth–Aaron pairs are:
|5]], 6), (24
24 (number)
24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...
, 25
25 (number)
25 is the natural number following 24 and preceding 26.-In mathematics:It is a square number, being 5² = 5 × 5. It is the smallest square that is also a sum of two squares: 25 = 3² + 4²...
), (49
49 (number)
49 is the natural number following 48 and preceding 50.-In mathematics:* Forty-nine is the square of seven and is therefore the fourth squared prime number.The aliquot sum of forty-nine is 8...
, 50
50 (number)
This article discusses the number fifty. For the year 50 CE, see 50. For other uses of 50, see 50 50 is the natural number following 49 and preceding 51.-In mathematics:...
), (77
77 (number)
77 is the natural number following 76 and preceding 78. Seventy-seven is the smallest positive integer requiring five syllables in English.-In mathematics:...
, 78
78 (number)
78 is the natural number following 77 and followed by 79.-In mathematics:78 is a triangular number, and its factorization makes it a sphenic number...
), (104
104 (number)
104 is the natural number following 103 and preceding 105.- In mathematics :One hundred [and] four is a primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable...
, 105
105 (number)
105 is the natural number following 104 and preceding 106.-In mathematics:105 is a triangular number, a 12-gonal number and a Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7...
), (153
153 (number)
One hundred fifty-three is the natural number following one hundred fifty-two and preceding one hundred fifty-four.-Mathematical properties:...
, 154
154 (number)
One hundred and fifty-four is the natural number following one hundred and fifty-three and preceding one hundred and fifty-five.-In mathematics:* 154 is a nonagonal number...
), (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:
, (8, 9), (15
15 (number)
15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name....
, 16
16 (number)
16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
), (77, 78), (125
125 (number)
125 is the natural number following 124 and preceding 126.-In mathematics:One hundred [and] twenty-five is the cube of 5....
, 126
126 (number)
126 is the natural number following 125 and preceding 127.-In mathematics:One hundred [and] twenty-six is a pentagonal pyramidal number, and a decagonal number as well as a pentatope number....
), (714, 715), (948, 949), (1330, 1331)
(The lesser of each pair is listed in ).
The intersection of the two lists begins:
, (77, 78), (714, 715), (5405, 5406)
(The lesser of each pair is listed in ).
Any Ruth-Aaron pair of square-free integer
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
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) = (7×112×19×443, 24×3×52×13×457). 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.
The name was given by Carl Pomerance
Carl Pomerance
Carl Bernard Pomerance is a well-known 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 has at least 7 distinct prime factors. He immediately joined the faculty at the...
for Babe Ruth
Babe Ruth
George Herman Ruth, Jr. , best known as "Babe" Ruth and nicknamed "the Bambino" and "the Sultan of Swat", was an American Major League baseball player from 1914–1935...
and Hank Aaron, as Ruth's career regular-season home run
Home run
In baseball, a home run is scored when the ball is hit in such a way that the batter is able to reach home safely in one play without any errors being committed by the defensive team in the process...
total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
at the University of Georgia
University of Georgia
The University of Georgia is a public research university located in Athens, Georgia, United States. Founded in 1785, it is the oldest and largest of the state's institutions of higher learning and is one of multiple schools to claim the title of the oldest public university in the United States...
at the time Aaron (a member of the nearby Atlanta Braves
Atlanta Braves
The Atlanta Braves are a professional baseball club based in Atlanta, Georgia. The Braves are a member of the Eastern Division of Major League Baseball's National League. The Braves have played in Turner Field since 1997....
) 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
only the 4 above triplets are known.