Problem 45994. Investigate the frequency of last digits of prime numbers

Created by ChrisR

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The last digit of a prime number greater than 5 can be 1, 3, 7, or 9. If the primes are distributed randomly, then these digits should be equally likely. However, <a href = "https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html">mathematicians discovered</a> relatively recently that--as Robert Lemke Oliver put it--these digits "really hate to repeat themselves". In other words, last digits repeat less often than expected.For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91), and none of them are followed by a prime ending in a 1. In fact, none of the primes less than 100 and ending in 3, 7, or 9 are followed by a prime ending in 3, 7, or 9, respectively.Write a function to compute the frequency of the last digits of primes between 7 and the input number. Return a matrix whose rows correspond to the digits of the first prime and columns correspond to the digits of the next prime. Please remember to (a) omit 2, 3, and 5 and (b) account for the prime following the last prime in your list. For the example given above, the function should return<pre> 0 0.6000 0.4000 0
 0 0 0.3333 0.6667
 0.6667 0.1667 0 0.1667
 0.4000 0.4000 0.2000 0</pre>For example, six primes less than 100 end in 7. They are followed by four primes ending in 1 (including 101), one prime ending in 3, zero primes ending in 7, and one prime ending 9. The frequencies, reported in the third row, are thus 2/3, 1/6, and 1/6.

The last digit of a prime number greater than 5 can be 1, 3, 7, or 9. If the primes are distributed randomly, then these digits should be equally likely. However, <https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html mathematicians discovered> relatively recently that--as Robert Lemke Oliver put it--these digits "really hate to repeat themselves". In other words, last digits repeat less often than expected.

For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91), and none of them are followed by a prime ending in a 1. In fact, none of the primes less than 100 and ending in 3, 7, or 9 are followed by a prime ending in 3, 7, or 9, respectively.

Write a function to compute the frequency of the last digits of primes between 7 and the input number. Return a matrix whose rows correspond to the digits of the first prime and columns correspond to the digits of the next prime. Please remember to (a) omit 2, 3, and 5 and (b) account for the prime following the last prime in your list. For the example given above, the function should return

0    0.6000    0.4000         0
         0         0    0.3333    0.6667
    0.6667    0.1667         0    0.1667
    0.4000    0.4000    0.2000         0

For example, six primes less than 100 end in 7. They are followed by four primes ending in 1 (including 101), one prime ending in 3, zero primes ending in 7, and one prime ending 9. The frequencies, reported in the third row, are thus 2/3, 1/6, and 1/6.

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Last Solution submitted on Mar 25, 2023

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Solution 10765934

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Christian Schröder on 26 Mar 2023 at 19:21

pivot() was introduced in R2023a --- this seemed like a good opportunity to use it.

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