To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Ruth–Aaron pair

From Wikipedia, the free encyclopedia

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.

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 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.[1]

YouTube Encyclopedic

  • 1/1
    Views:
    459
  • ✪ The Number 153 : more here: http://goo.gl/9aKzQ

Transcription

Contents

Examples

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 OEISA006145).

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 OEISA039752).

The intersection of the two lists begins:

(5, 6), (77, 78), (714, 715), (5405, 5406).

(The lesser of each pair is listed in OEISA039753).

Any Ruth–Aaron pair of square-free integers 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.

Density

Ruth-Aaron pairs are sparse (that is, they have density 0). This was conjectured by Nelson et al. in 1974[2] and proven in 1978 by Erdős and Pomerance.[3]

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.[citation needed]

See also

References

  1. ^ Aaron Numbers -- Numberphile
  2. ^ Nelson, C.; Penney, D. E.; and Pomerance, C. "714 and 715." J. Recr. Math. 7, 87-89, 1974.
  3. ^ Erdős, P. and Pomerance, C. "On the Largest Prime Factors of n and n+1." Aequationes Mathematicae 17, 311-321, 1978.

External links

  • Weisstein, Eric W. "Ruth-Aaron pair". MathWorld.
  • "Ruth–Aaron Triplets" and "Ruth–Aaron pairs revisited". The prime puzzles & problems connection. Retrieved November 9, 2006.
This page was last edited on 8 January 2019, at 20:56
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.