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i need to know the maximum possible number of prime numbers between $10n$ and $10n+10$ for $n>0$.

So far I've found the only possible primes are $10n+1$, $10n+3$, $10n+7$, and $10n+9$ but there are values of n where some of these are not prime. I've got no clue on how to go beyond this so I'll appreciate some help here.

CoolJedi132
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You don't need every value of $n$ to have the property that those numbers are prime, only one such is needed. And, why, if you take $n = 1$...

Duncan Ramage
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  • Oh dear, i can't believe i didn't try with 1 before, thank you. Well, no that i posted, is there another way to get to that conclusion? – CoolJedi132 Oct 23 '20 at 02:40
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    @CoolJedi132 Perhaps; it's awfully hard to say conclusively that a proof doesn't exist. However, your solution is a demonstration of a very common method of proving "the maximum number of things is...". First, show that there is some upper bound (in your case, 4), and then show that this bound is achieved. – Duncan Ramage Oct 23 '20 at 03:04
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    @CoolJedi132 $n=10, 19, $ also works. See OEIS – Calvin Lin Oct 23 '20 at 03:21