Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis

[Joe Horn]

(08-06-2017 09:47 AM)Gerald H Wrote:  To avoid suspense: Horn's Conjecture is WRONG. ... I arrived at a heuristic proof.

What is the probability of there being exactly one cube, call it H, with decimal digits exclusively 2,3,5,7 & each of these digits appearing in the representation?

Difficult to say, but surely very small. [That's not zero. -jkh-]

The probability can be further diminished [to zero? -jkh-] by adding that Joe found H.

That Joe found H amongst the infinitude of cubes can only indicate that there must be [WRONG -jkh-] a large number of cubes with the required property. [No, it indicates that there MIGHT be MORE THAN ONE. That's all. -jkh-]

Indeed, using the maximum likelihood hypothesis, the greatest probability of Joe finding H occurs when there is an infinity of such cubes. [That makes it more likely, but it is not necessary. In other words, if my hypothesis were correct, would that PREVENT me from finding H? Of course not. -jkh-]

Proving [WRONG. -jkh-], to all intents & purposes [AHA! There it is. -jkh-], that there is an infinity of the proposed cubes.

QED [ <-- Too soon. -jkh-]

See my comments inserted above, inside [ square ] brackets.

As mentioned in the original posting, a counterexample would suffice to disprove the conjecture. The above rationale explains why the conjecture is probably false, but it does not disprove it.

FWIW, if the above were a valid proof, then it would also prove the existence of infinitely many perfect square repunits, since 1 itself is certainly one such. But there are no others. Therefore the "proof" is invalid. QED.