Cody

Problem 46087. Investigate the frequency of last non-zero digits of primorials

Cody Problems 46054, 46057, and 46060 involve the primorial p_n#, or the product of prime numbers up to and including the nth prime p_n. In particular, the second of these deals with the last non-zero digit (LNZD) in the primorial. For example, the first eleven primorials are
2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230
After the second primorial, the LNZD is 1, 3, 7, or 9. This observation holds for larger n as well because prime numbers greater than 5 end in 1, 3, 7, or 9 and products of numbers whose LNZDs are 1, 3, 7, or 9 also have a LNZD of 1, 3, 7, or 9.
Mathematicians have determined that the last digits of prime numbers repeat themselves less often than expected. If these digits occurred randomly, 25% of the primes that follow primes ending in a 1 should also end in a 1. However, as verified in Cody Problem 45994, for primes up to 100 million, the actual frequency is less than 18%.
What about primorials? What happens when basic multiplication facts interact with the frequency of last digits of primes? Will the frequencies be similarly skewed? Or will they all approach 25%?
Write a function that takes as input an integer n and computes the frequency of the last digits of primorials between p_3# = 30 and p_n#. Return a matrix whose rows correspond to the digits of the first primorial and columns correspond to the digits of the next primorial. Please remember to (a) omit p_1# = 2 and p_2# = 6 and (b) account for the primorial following p_n#. For example, if your function is given n = 10, then it should return
0.3333 0.3333 0 0.3333
0.6667 0.3333 0 0
0 1.0000 0 0
0 0 1.0000 0
That is, with n = 10, we consider eight primorials (p_3# through p_10#). Of these eight, three have a LNZD of 3 (30, 30030, and 6469693230). Two of those (i.e., 66.7%) are followed by primorials with a LNZD of 1, and one (33.3%) is followed by a primorial with a LNZD of 3.
Solve

Solution Stats

90.91% Correct | 9.09% Incorrect
Last Solution submitted on Apr 30, 2023

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