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Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis
08-06-2017, 09:47 AM
Post: #6
RE: Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis
To avoid suspense: Horn's Conjecture is WRONG.

After prolonged hesitation & cogitation I decided to join the throng of adepts attempting to solve the question of Joe's conjecture.

First some general observations:

1 The ancient Greeks have very little to say: they were not interested in digits in a positional representation of numbers, they were interested in the properties of numbers;

2 The proof below exceeds the limits of the margin, nevertheless I will publish the complete proof to expose it to the court of my peers, superiors & inferiors;

3 I am not interested in financial gain.

An approach is to convert a problem in multiplicative number theory (structure of x^3) to one of additive number theory (sum of consecutive odd integers).

For integer input N the programme below returns N & a list of consecutive odd integers that sum to n^3:

    FPTR2 ^#>Z
    FPTR2 ^#>Z
However, having considered the bijection of cubes & sums of consecutive odd integers I concluded this was a dead-end.

& similarly hopeless is a proof in the traditional sense, as base 10 representation of a number tells us more about 10 than of the number represented. Number theory speaks of the properties of numbers, eg

3153023022 base 7 is an odd number,

& this remains true if converted to base 10 or any base.

So 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.

The probability can be further diminished by adding that Joe found H.

That Joe found H amongst the infinitude of cubes can only indicate that there must be a large number of cubes with the required property.

Indeed, using the maximum likelihood hypothesis, the greatest probability of Joe finding H occurs when there is an infinity of such cubes.

Proving, to all intents & purposes, that there is an infinity of the proposed cubes.

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RE: Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis - Gerald H - 08-06-2017 09:47 AM

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