I was surprised to find that my diurnal age today, 26914, is part of a triple of numbers that almost satisfies the equation
A050792 | Consider the Diophantine equation |
The initial values are: 9, 64, 73, 135, 334, 244, 368, 1033, 1010, 577, 3097, 3753, 1126, 4083, 5856, 3987, 1945, 11161, 13294, 3088, 10876, 16617, 4609, 27238, 5700, 27784, 11767, 26914.
The corresponding values are given by OEIS A050791:
A050791 | Consider the Diophantine equation |
The initial values are: 12, 103, 150, 249, 495, 738, 1544, 1852, 1988, 2316, 4184, 5262, 5640, 8657, 9791, 9953, 11682, 14258, 21279, 21630, 31615, 36620, 36888, 38599, 38823, 40362, 41485, 47584.
Given the and values, we can calculate the corresponding values and so the initial triples that satisfy are:
- (9, 10, 12)
- (64, 94, 103)
- (73, 144, 150)
- (135, 235, 249)
- (334, 438, 495)
- (244, 729, 738)
- (368, 1537, 1544)
- (1033, 1738, 1852)
- (1010, 1897, 1988)
- (577, 2304, 2316)
- (3097, 3518, 4184)
- (3753, 4528, 5262)
- (1126, 5625, 5640)
- (4083, 8343, 8657)
- (5856, 9036, 9791)
- (3987, 9735, 9953)
- (1945, 11664, 11682)
- (11161, 11468, 14258)
- (13294, 19386, 21279)
- (3088, 21609, 21630)
- (10876, 31180, 31615)
- (16617, 35442, 36620)
- (4609, 36864, 36888)
- (27238, 33412, 38599)
- (5700, 38782, 38823)
- (27784, 35385, 40362)
- (11767, 41167, 41485)
- (26914, 44521, 47584)
Any number of solutions to the equationHere are the triples thrown up by the parameterized solutions for values ofcan be produced through the use of the algebraic identity by substituting in values of . Although these are certainly solutions, the identity generates only one family of solutions. Other solutions such as (64, 94, 103), (135, 235, 249) and (334, 438, 495) can be found. What is not known is if it is possible to parameterize all solutions for this equation. Put another way, are there an infinite number of families of solutions? Probable yes, but that too remains to be shown.
- [9, 10, 12]
- [73, 144, 150]
- [244, 729, 738]
- [577, 2304, 2316]
- [1126, 5625, 5640]
- [1945, 11664, 11682]
- [3088, 21609, 21630]
- [4609, 36864, 36888]
- [6562, 59049, 59076]
- [9001, 90000, 90030]
- [11980, 131769, 131802]
- [15553, 186624, 186660]
- [19774, 257049, 257088]
- [24697, 345744, 345786]
- [30376, 455625, 455670]
- [36865, 589824, 589872]
- [44218, 751689, 751740]
- [52489, 944784, 944838]
- [61732, 1172889, 1172946]
- [72001, 1440000, 1440060]
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