Prime numbers can end in 1,3,5, 7, 9.
I wonder, which ending digit has the most abundant prime numbers, is there any proof?
Asked
Active
Viewed 669 times
0
-
Maybe there is a relation with Benford's law https://en.wikipedia.org/wiki/Benford%27s_law. – Apr 20 '20 at 06:23
-
1They had a look at this question in numberphile with this video: https://www.youtube.com/watch?v=YVvfY_lFUZ8 – JC12 Apr 20 '20 at 06:29
-
2"Prime numbers can end in 1,3,5, 7, 9." -- If you're going to include $5$, you might as well include $2$. $2$ and $5$ are the only primes ending in those digits though. – PrincessEev Apr 20 '20 at 06:35
-
1Dirichlet's Theorem tells us that each of the four cases $1,3,7,9$ are equally probable on average. Is that what you are after? – lulu Apr 20 '20 at 10:37
-
@lulu Yes, this is exactly what I want to know... – an offer can't refuse Apr 21 '20 at 09:46
1 Answers
1
You are after Chebychev's bias. Squares are slightly less likely than non-squares. Which is to say, primes ending in $3$ and $7$ are slightly more common than primes ending in $1$ and $9$. Any actual proof of this fact (at least in general) requires at least the Riemann hypothesis, as far as Wikipedia can tell.
However, the bias is small. If you look at the ratio of the number of primes that are congruent to $1$ compared to the number of primes that are congruent to $3$ (rather than the difference that Chebychev's bias does), this ratio tends to $1$. Here is a short proof.
Primes ending with even numbers and with $5$ are so rare we don't even bother including them in this context.
Arthur
- 199,419