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A261296 Smaller of pairs (m, n), such that the difference of their squares is a cube and the difference of their cubes is a square.

%I A261296 #27 Sep 19 2024 19:39:44
%S A261296 6,384,4374,5687,24576,17576,27783,64350,93750,354375,279936,113750,
%T A261296 363968,166972,370656,705894,263736,1572864,1124864,1778112,3187744,
%U A261296 4225760,4118400,3795000,3188646,4145823,4697550,1111158,730575,6000000,8171316,2413071,8573750
%N A261296 Smaller of pairs (m, n), such that the difference of their squares is a cube and the difference of their cubes is a square.
%C A261296 The numbers come in pairs: (6,10), (384, 640) etc. The larger numbers of the pairs can be found in A261328. The sequence has infinite subsequences: Once a pair is in the sequence all its zenzicubic multiples (i.e., by a 6th power) are also in this sequence. Primitive solutions are (6,10), (5687, 8954), (27883, 55566), (64350, 70434), ....
%C A261296 Assumes m, n > 0 as otherwise (k^6, 0) will be a solution. Sequence sorted by increasing order of largest number in pair (A261328). - _Chai Wah Wu_, Aug 17 2015
%D A261296 H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177
%H A261296 Chai Wah Wu, <a href="/A261296/b261296.txt">Table of n, a(n) for n = 1..302</a>
%H A261296 Gianlino, in reply to Smci, <a href="https://answers.yahoo.com/question/index?qid=20110722023859AAsGZxn">Solution method for "integers with the difference between their cubes is a square, and v.v."</a>, Yahoo! answers, 2011
%e A261296 10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.
%e A261296 8954^3 - 5687^3 = 730719^2, 8954^2 - 5687^2 = 363^3.
%o A261296 (Python)
%o A261296 def cube(z, p):
%o A261296     iscube=False
%o A261296     y=int(pow(z, 1/p)+0.01)
%o A261296     if y**p==z:
%o A261296         iscube=True
%o A261296     return iscube
%o A261296 for n in range (1, 10**5):
%o A261296     for m in range(n+1, 10**5):
%o A261296         a=(m-n)*(m**2+m*n+n**2)
%o A261296         b=(m-n)*(m+n)
%o A261296         if cube(a, 2)==True and cube(b, 3)==True:
%o A261296             print (n, m)
%Y A261296 Cf. A000290, A000578, A001014, A261328.
%K A261296 nonn
%O A261296 1,1
%A A261296 _Pieter Post_, Aug 14 2015
%E A261296 Added a(6) and more terms from _Chai Wah Wu_, Aug 17 2015