Fun with Numbers: The Pan-Prime-Digit Cube Hypothesis
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08-06-2017, 09:47 AM
Post: #6
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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: Code:
& 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. QED |
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