Properties
All strictly non-palindromic numbers beyond 6 are prime. To see why composite n > 6 cannot be strictly non-palindromic, for each such n a base b must be shown to exist where n is palindromic.
- If n is even, then n is written 22 (a palindrome) in base b = n/2 − 1.
Otherwise n is odd. Write n = p · m, where p is the smallest prime factor of n. Then clearly p ≤ m.
- If p = m = 3, then n = 9 is written 1001 (a palindrome) in base b = 2.
- If p = m > 3, then n is written 121 (a palindrome) in base b = p − 1.
Otherwise p < m − 1. The case p = m − 1 cannot occur because both p and m are odd.
- Then n is written pp (the two-digit number with each digit equal to p, a palindrome) in base b = m − 1.
The reader can easily verify that in each case (1) the base b is in the range 2 ≤ b ≤ n − 2, and (2) the digits ai of each palindrome are in the range 0 ≤ ai < b, given that n > 6. These conditions may fail if n ≤ 6, which explains why the non-prime numbers 1, 4 and 6 are strictly non-palindromic nevertheless.
Therefore, all strictly non-palindromic n > 6 are prime.
Read more about this topic: Strictly Non-palindromic Number
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