Ruth-Aaron Pairs Trivia Quiz

Answer: (77,78)

The sum of prime factors is also known as prime divisor sum.

Only (77,78) qualifies as a Ruth-Aaron pair:

77=7×11, 78=2×3×13 both with a prime divisor sum of 18 (7+11=2+3+13=18).

The rest of the pairs fail to meet the criterion, for example (93,94):

93=3×31 with a prime divisor sum of 34 (3+31).
94=2×47 with a prime divisor sum of 49 (2+47).

Answer: Both

Both pairs qualify:

(8,9):
8=2×2×2, 9=3×3 both with a prime divisor sum of 6 (2+2+2=3+3=6).

(15,16):
15=3×5, 16=2×2×2×2 both with a prime divisor sum of 8 (3+5=2+2+2+2=8).

Answer: The number 714715 is a perfect square

714715 is not a perfect square. A simple rule of thumb says that all perfect squares which end in the digit 5, also end in the digits 25.

(714,715) was the first pair to be discussed and also gave Ruth-Aaron pairs their name:

714=2×3×7×17, 715=5×11×13 both with a prime divisor sum of 29.

Indeed, the above factorizations use all prime numbers up to 17 (2,3,5,7,11,13 and 17).

714+715=1429 is a prime number, as are a few of its permutations: 9241, 1249, 9421, 4129 and 4219.

The three correct statements, and more, are introduced in the initial publication on Ruth-Aaron pairs by Pomerance et al. (http://www.trottermath.net/numthry/ruth714.html).

Answer: Baseball legends Babe Ruth and Hank Aaron

On April 8, 1974 Hank Aaron hit his 715th home run, breaking Babe Ruth's home run record of 714. The ecstasy which engulfed the United States did not skip the department of mathematics in the University of Georgia. Number theorist Dr. Carl Pomerance and his peers preoccupied themselves with thoughts on these numbers, and it was a student named Jeremy Jordan who noticed that the prime divisor sum of 714 and 715 is identical. Thus opened another chapter in the history of number theory.

A reference with lots of additional information would be the lecture given by Carl Pomerance in the 2002 Paul Erdõs memorial seminar (http://math.dartmouth.edu/~carlp/PDF/paper130.pdf).

Answer: Paul Erdõs

Paul Erdõs was one of the most prolific mathematicians of the 20th century. His eccentric mathematical devotion and genius are best described by the title of his biography - "The Man Who Loved Only Numbers" (Paul Hoffman, 1998). Erdõs read the article by Pomerance et al. and contacted Pomerance letting him know that he can prove that Ruth-Aaron numbers are of density 0 (i.e. very sparse). This started a long-time collaboration between these two mathematicians.

In 1995, Emory University granted an honorary degree to both Paul Erdõs and Hank Aaron. It is said that the baseball they signed at the event gave Aaron what many a mathematician would kill to get - an "Erdõs number" of 1.

The other Hungarians mentioned in the question were busy doing other things in 1974: Architect Ernõ Rubik must have started contemplating his cube, football great Ferenc Puskás managed Greek club Panathinaikos and horror actor Béla Lugosi was long gone to the underworld.

Answer: 20

Ruth-Aaron pair lists constructed by Ted Alper can be found at http://www.trottermath.net/numthry/ravdata.html

Up to 1 million, Alper lists 149 pairs.

The last achievement in this area was made by Carl Pomerance (2002) who showed that the number of Ruth-Aaron pairs up to N is of O(N/lnN).

Answer: False

Lies, lies and lies again. Ruth-Aaron pairs were only "discovered" in 1974. As of 2009 the infinity of Ruth-Aaron pairs still awaits proof. The Israeli society, as does the methematical community, are yet to be blessed by the genius of one Aharon Ruthfeld. I beg your forgiveness.

For an updated status on the proof, please refer to the online Bible of mathematics at http://mathworld.wolfram.com/Ruth-AaronPair.html

Interestingly, Joe K. Crump managed to devise an algorithm which can produce arbitrarily big Ruth-Aaron pairs: http://www.immortaltheory.com/NumberTheory/RuthAaron.htm

Answer: 1

And the winner is 5, the first member of the first Ruth-Aaron pair (5,6).

By definition, the prime divisor sum of a prime number is the prime number itself, which intuitively implies that there are no other prime Ruth-Aaron numbers since as we progress on the scale of natural numbers, the prime divisor sum of a non-prime number will always be much smaller than the number itself.

Answer: Yes

Very few Ruth-Aaron triplets have been found to date.

Here is the smallest one:

417162=2×3×251×277, 417163=17×53×463, 417164=2×2×11×19×499 all with a prime divisor sum of 533.

The rest of the known Hank-Aaron triplets can be found at http://en.wikipedia.org/wiki/Ruth-Aaron_pair

Answer: 40

8281=91×91=7×7×13×13, 8280=2×2×2×3×3×5×23 both with a prime divisor sum of 40.

This quiz is dedicated to my wife on her birthday, hope you enjoyed it. Being a gentleman as well as a gentlegiant, I will not disclose her name or age here (did I hear someone say "Easter eggs"?). Mazal Tov & till 120!