Consider positive integers which are perfect cubes, do not end in zero, are not palindromes, and their reverse is also a perfect cube.

Q1: Find the smallest such reversible cube which also contains a string of 4 consecutive zeros.

Q2: What do the following pairs of integers have to do with these reversible cubes, and what pair would come next? (hint: see Conjecture 1)

(11, 13), (19, 25), (35, 49), (37, 41), (39, 57), (43, 53), (67, 97)

Conjectures:

(1) the cube roots of all reversible cubes contain only two unique digits, 'a' and 'b'.

(2) the first digit of all reversible cubes is 'c' and the next non-zero digit is always 'd'.

Bonus Questions related to Conjectures:

What are a and b; and what are c and d?

Can you prove the conjectures to be either true or false? (I have not been able to do so)

If a proof is not possible, can you offer an explanation for this finding? (I cannot)

(In reply to computer exploration by Charlie)

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Posted by Mark on 2021-09-30 04:44:12