Number Choosing Contest.

It's worth noting that -4 also passed rule 5, though to be honest given how out of my way I went to get 331130809 I think "perfect squares, bar possible mistakes" is the de facto rule 5.

Rule 3, if you want to spoil yourself:

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SpoilerMust be a binary palindrome (e.g. 85 = 1010101 or 331130809 = 10011101111001010011110111001)

(this is why I had to get Python involved; to find numbers that pass both rules 3 and 5)

(yes I did get that from just 0, 1, 3, 5, 7, 9 & 85)

(sorry)