For a prime $p>2,$ the only possible final digits are 1,3,7,9. My question is: taking any prime $p>2$ at random, what are the probabilities that it ends in each of the given numbers. Any references/links to relevant literature would be appreciated.
What are the probabilities for the last digits of prime numbers?
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probability
number-theory
elementary-number-theory
prime-numbers
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8By [Dirichlet's Theorem](https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions) the answer is $\frac 1{\varphi(10)}=\frac 14$ for each. – 2017-02-27
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1You can't take any prime at random, much like you can't take any number at random. – 2017-02-27
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2@IvanNeretin Good discussions of Dirichlet's Theorem should go into the definition of the probability involved. Usually, the theorem works with "analytic density" based on sums of the reciprocals of primes in the relevant progression. That is unintuitive, but easier to compute with. The theorem is also true for "natural density" , wherein you compute below some cap $N$ and then let $N$ grow to infinity though people seldom refer to that and the proof is much less well known. – 2017-02-27
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1What if I go to a website like Random.org, it gives a number in a certain given range, like $298$, then I look up the prime number by that index, e.g., $1973$? – 2017-02-27