Primes that don’t look like primes

Primes usually look odd. They’re literally odd [1], but they typically don’t look like they have a pattern, because a pattern would often imply a way to factor the number.

However, 12345678910987654321 is prime!

I posted this on Twitter, and someone with the handle lagomoof replied that the analogous numbers are true for bases 2, 3, 4, 6, 9, 10, 16, and 40. So, for example, the hexadecimal number 123456789abcdef10fedcba987654321 would be prime. The following Python code proves this is true.

    from sympy import isprime

    for b in range(2, 41):
        n = 0
        for i in range(0, b):
            n += (i+1)*b**i
        for i in range(b+1, 2*b):
            n += (2*b-i)*b**i
        print(b, isprime(n))

By the way, some have suggested that these numbers are palindromes. They’re not quite because of the 10 in the middle. You can show that the only prime palindrome with an even number of digits is 11. This is an example of how a pattern in the digits leads to a way to factor the number.

Related posts

[1] “All primes are odd except 2, which is the oddest of all.” — Concrete Mathematics

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