Back to . . . .  NCB Deposit  # 95 Jorge Robeiro Department of English CSULA jorgeribeiro28@mac.com Number Theory, Baseball and Fiction  Ruth-Aaron Pairs: Prime Number Theory From ( IX, 20) of the Elements.

Are there an infinite number of Ruth-Aaron numbers?

Paul Erdös (1913-1996) was one of the most prolific mathematicians of all time, having authored over 1500 articles. An Erdös number measures the shortest distance of a researcher from Erdös on a graph whose edges represent co-authorship.

The NCB thanks number theorist Chris Caldwell, University of Tennessee, Martin,
for much of the above information.  Carl Pomerance is now at Dartmouth.

Some numbers . . . .

Ruth-Aaron numbers are a pair of consecutive integers such that the sum of the prime factors of  n and  n + 1 are equal.

Examples:
 Counting distinct prime factors only once: Prime Factors: Sum of Prime Factors: 5   and  6 5 2, 3 5  =  5 2  +  3  =  5 24   and  25 2, 3 5 2  +  3  =  5 5  =  5 49  and  50 7 2, 5 7  =  7 2 + 5  =  7 77  and  78 7, 11 2, 3, 13 7 +  11  =  18 2 +3 +13  =  18 104  and  105 2, 13 3, 5, 7 2 +  13  =  15   3 + 5 + 7  =  15 153  and  154 3, 17 2, 7, 11 3 +  17  =  20   2 + 7 +11  =  20 369  and  370 3, 41 2, 5, 37 3 +  41  =  44 2 + 5 + 37  =  44 492  and  493 2, 3, 41 17, 29 2 + 3  +  41  =  46   17 +  29  =  46 714  and  715 2, 3, 7, 17 5, 11, 13 2+3 +7 +17 =  29    5 + 11 + 13  = 29 and for more examples . . . .

 and with repeated prime factors - - - more Ruth-Aaron pairs 8  and  9 2, 2, 2, 3, 3 2 + 2 + 2  =  6 3 + 3  =  6 15  and  16 3, 5 2, 2, 2, 2 3 + 5  =  8 2 + 2 + 2 + 2  =  8 Ruth-Aaron triplets when counting repeated prime factors 41762 41763 41764 2,  3,  251,  277 17, 53, 463 2, 2, 11, 19, 499 2 +3 + 251 + 277 = 533 17 + 53 + 463 = 533 2 + 2 + 11 + 19 + 499= 533

References

 P. Erdös and C. Pomerance, On the largest prime factors of n and n + 1, Aequations Mathematicae 17, (1978, 311-321. C. Nelson, D. E. Penny and C. Pomerance, 714 and 715,  J. Recreational Math., 7, (1974), 87-89. C. Pomerance, Ruth-Aaron Numers Revisited, Paul Erdös and His Mathematic I, Budapest 2002, pp. 567-579, Mathematical Studies, 11, Bolyai Society. A. Schnizel and W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers. Remarque, Acta Arithm. 4, 185-208, 1958.

 Significance of Deposit #95:  "The Housekeeper and the Professor" Obviously, many people of all ages and cultures have come to appreciate statistics through a love of the game of baseball.  In a very subtle manner the Japanese writer Yoko Ogawa uses baseball, number theory and struggles in life to form a common bond among three individuals - a memory impaired professor, a housekeeper and the housekeeper's illegitimate ten-year-old son.   There are numerical bonds other than Ruth-Aaron pairs.  To quote Katharine Ott's review in Science (vol. 324, June 5, 2009), Ogawa's narrative "blossoms in the passages describing the Housekeeper's journey into mathematics."  Ogawa uses birthdays to remind math readers of the Pythagorean/Euclidean interest in amicable numbers.  She links human traits and emotions with perfect, deficient and abundant numbers. She indirectly reminds readers of basic questions often attributed to Descartes.  "Math has proven the existence of God because it is absolute and without contradiction; but the devil must exist as well, because we cannot prove it."

 and for all primes . . . Prime Number Graphs:  Study the above two curves.  On the left, the number of primes between 0 and 100 would appear to be "locally" irregular.  When the scale is increased on the right graph, the curve would appear to be far smoother and far more predictable.

For several years Chris K. Caldwell of the University of Tennessee, Martin, has created and maintained an indispensable web site on prime number theory.  The NCB was lucky to have had his timely Deposit # 19 on twin primes. organizer George Woltman. He reported the forty-second Mersenne prime with the work being verified by Tony Reix.  This prime has 7,816,230 digits, the largest of any type, and may be expressed as

 The NCB recommends the following sources . . . . . . on Primes and Mersenne Primes
How Many Primes Are There?  See  < http://primes.utm.edu/howmany.shtml  >
Frequently Asked Questions about Prime Numbers.  See  <  http://primes.utm.edu/notes/faq/  >
What are the first twenty largest known  "twin primes?"  See  < http://www.research/primes/largest.html#twin  >
See GIMPS' <  http://www.mersenne.org  >  and Focus, The Newsletter of the Mathematical Association of America, vol. 24, February, 2004, p. 4.
Jeffrey J. Wanko, "The Legacy of Marin Mersenne," Mathematics Teacher, vol. 98, No. 8, April, 2005, pp. 525-529.
and from the BBC News on April 4, 2003,  see  <  http://news.bbc.co.uk/2/hi/science/nature/2911945.stm  >
 . . . and on Ruth-Aaron pairs.
What are the ten largest known  "Ruth-Aaron" pairs?  See  < http://www.research.att.com/~njas/sequences/A006145  >
What are "Ruth-Aaron" triplets?  See  < http://en.wikipedia.org/wiki/Ruth-Aaron_pair >
What is in the research literature?  See  < http://mathworld.wolfram.com/Ruth-AaronPair.html >

Books you will enjoy . . . .

 "And when he (the Professor) noticed that his seat number was 714 and Root's was 715, he began to lecture again and completely forgot to sit down." Yoko Ogawa in her widely acclaimed The Housekeeper and the Professor," Picador, 2009. (Translated from the Japanese.)
 "There's no such thing as a number devil." "Is that so?  How can you be speaking to me if I don't exist?" Hans Magnus Enzensberger in his best selling   The Number Devil:  A Mathematical Adventure.  Metropolitan Books, Hold and Co., 1997. (Translated from the German.)