Problem 45994. Investigate the frequency of last digits of prime numbers
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, 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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The comment "For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91)" should read "For example, five primes less than 100 end in a 1 (11, 31, 41, 61, and 71)". 91 is not a prime.
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