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Smaller of pairs (m, n), such that the difference of their squares is a cube and the difference of their cubes is a square.
2

%I #27 Sep 19 2024 19:39:44

%S 6,384,4374,5687,24576,17576,27783,64350,93750,354375,279936,113750,

%T 363968,166972,370656,705894,263736,1572864,1124864,1778112,3187744,

%U 4225760,4118400,3795000,3188646,4145823,4697550,1111158,730575,6000000,8171316,2413071,8573750

%N 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 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 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 H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177

%H Chai Wah Wu, <a href="/A261296/b261296.txt">Table of n, a(n) for n = 1..302</a>

%H 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 10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.

%e 8954^3 - 5687^3 = 730719^2, 8954^2 - 5687^2 = 363^3.

%o (Python)

%o def cube(z, p):

%o iscube=False

%o y=int(pow(z, 1/p)+0.01)

%o if y**p==z:

%o iscube=True

%o return iscube

%o for n in range (1, 10**5):

%o for m in range(n+1, 10**5):

%o a=(m-n)*(m**2+m*n+n**2)

%o b=(m-n)*(m+n)

%o if cube(a, 2)==True and cube(b, 3)==True:

%o print (n, m)

%Y Cf. A000290, A000578, A001014, A261328.

%K nonn

%O 1,1

%A _Pieter Post_, Aug 14 2015

%E Added a(6) and more terms from _Chai Wah Wu_, Aug 17 2015