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How can I prove that the last digit of a prime number other than $ 2 $ and $ 5 $ can be $ 1,3,7,9 $ without using Dirichlet's theorem?

It occurs to me to try to prove that

$ 4n + 1 \equiv a_n \pmod {10} $

$ 4n + 3 \equiv a_n \pmod {10} $

with $ a_n = \{1,3,7,9 \} $

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    $\begingroup$ As stated, you just need to exhibit primes with those last digits, like $11,13,7,19$ and you are done. I believe you mean to ask how to prove that all primes other than $2$ and $5$ end in one of those digits. $\endgroup$ – Ross Millikan Oct 31 at 4:21
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If the last digit is $2,4,6,8,0$ the number is even. If the last digit is $5$ the number is divisible by $5$. Thus, if the number is prime it can only end in $1,3,7,9$.

In general, primes in base $n$ that are greater than $n$ can only have last digits coprime to $n$.

Specific primes that end in $1,3,7,9$ are $11,13,17$ and $19$.

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    $\begingroup$ And in base 2 the last digit can only be 1. So generally in base n, the last digit(s) can only be... what subset of {0,...,n-1}? $\endgroup$ – John Forkosh Oct 31 at 4:10
  • $\begingroup$ (at)ParclyTaxel for anybody. What's the function $f:\mathbb N \to 2^{\mathbb N}$ such that $f(n)\subseteq\{0,\ldots,n-1\}$ is the appropriate subset? $\endgroup$ – John Forkosh Oct 31 at 4:35
  • $\begingroup$ (at)ParclyTaxel Thanks. $\endgroup$ – John Forkosh Oct 31 at 4:42

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