Drop links or images here to add them to the editor.

The legendary baseball player Babe Ruth hit an unseen lifetime record of 714 home runs. This record stood until April 8, 1974, when it was surpassed by Hank Aaron's record-breaking 715th home run. Carl Pomerance — mathematician at the University of Georgia at the time Aaron broke Ruth's record — introduced a new concept in mathematics based on this event. The inspiration came when a student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 are equal. He therefore defined a Ruth-Aaron pair as a pair of consecutive natural numbers $$(n, n + 1)$$ such that the sums of the prime factors of $$n$$ and $$n + 1$$ are equal.

Each positive integer $$n \in \mathbb{N_0}$$ can be written as a product of prime numbers. \[\begin{eqnarray}10 &=& 2 \times 5\\12 &=& 2 \times 2 \times 3\\13 &=& 13\\100 &=& 2 \times 2 \times 5 \times 5 \end{eqnarray}\] This decomposition in prime factors is unique, except for the order of the prime factors. The pair $$(714, 715)$$ is a Ruth-Aaron pair as \[\begin{eqnarray}714 &=& 2 \times 3 \times 7 \times 17\\715 &=& 5 \times 11 \times 13\end{eqnarray}\] and $$2 + 3 + 7 + 17 = 29 = 5 + 11 + 13$$. The pair $$(9, 10)$$ is not a Ruth-Aaron pair as $$2 + 5 \not= 3 + 3$$.

Assignment

Write a function isPrime that takes a positive integer $$n \in \mathbb{N_0}$$. The function must return a Boolean that indicates whether or not $$n$$ is prime.
Write a function primeFactorsthat takes a positive integer $$n \in \mathbb{N_0}$$. The function must return the list of prime factors of $$n$$, sorted in increasing order.
Write a function isRuthAaron that takes two positive integers $$m, n \in \mathbb{N_0}$$. The function must return a Boolean that indicates whether or not the pair $$(m, n)$$ is a Ruth-Aaron pair. Note that $$(m, n)$$ can only be a Ruth-Aaron pair if the integer $$n$$ immediately follows the integer $$m$$. The pair $$(714, 715)$$ is a Ruth-Aaron pair, but the pair $$(715, 714)$$ isn't.

Example

>>> isPrime(2)
True
>>> isPrime(6)
False
>>> isPrime(12)
False

>>> primeFactors(12)
[2, 2, 3]
>>> primeFactors(17)
[17]
>>> primeFactors(18)
[2, 3, 3]

>>> isRuthAaron(5, 6)
True
>>> isRuthAaron(10, 11)
False
>>> isRuthAaron(15, 16)
True
>>> isRuthAaron(8281, 8280)
False

Resources

Babai L, Pomerance C, Vértesi P (1998). The Mathematics of Paul Erdos. Notices Amer. Math. Soc. 45, 19-23.
Drost JL (1997). Ruth/Aaron Pairs. J. Recr. Math. 28(2), 120-122.
Erdos P, Pomerance C (1978). On the Largest Prime Factors of $$n$$ and $$n+1$$. Aeq. Math. 17, 311-321.
Hoffman P (1998). The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion.
Mackenzie D (1997). Mathematics: Homage to an Itinerant Master. Science 275, 759.
Nelson C, Penney DE, Pomerance C (1974). 714 and 715. J. Recr. Math. 7, 87-89.
Peterson I (2005). MathTrek: Playing with Ruth-Aaron Pairs. Sci. News 168, 2005.
Pomerance C (2002). Ruth-Aaron Numbers Revisited. In Paul Erdos and his Mathematics. I. Papers from the Conference Held in Budapest, July 4-11, 1999 (Ed. G. Halász, L. Lovász, M. Simonovits, and V. T. Sós). Berlin: Springer-Verlag, 567-579.