## 20062Re: 2^m+3^n and 2^n+3^m

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• Apr 11 6:48 AM

> "Numbers n such that there are no primes of the forms
> 2^m+3^n or 2^n+3^m for m < n"

These numbers include

1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 7251, 8218

but my coverage of the range n < 8218 is not complete,
so this is not yet a bona fide sequence.

So far, I have tested more than 7000 values of n < 8300
and found these 10 examples.

This agrees tolerably well with my prior heuristics.

To make a rough estimate of the density of primes of the
form N = 2^a + 3^b, I sieved for prime factors < 10^5,
with a and b running from 5000 to 5200, and found that
5558 values of N survived. So I estimated the
probability of primality to be C/log(N), with

C =~ exp(Euler)*log(10^5)*5558/201^2 =~ 2.82.

Hence I guessed that the probability of finding that
a number n lies in this sequence exceeds

exp(-C/log(2))*exp(-C/log(3)) > 1/800.

I say "exceeds", since 2^n+3^m starts at size
O(2^n) and ends at size O(3^n), as m runs from
1 to n-1, while 2^m+3^n stays at size O(3^n).

David
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