OFFSET
1,1
COMMENTS
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), ....
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
REFERENCES
H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..302
Gianlino, in reply to Smci, Solution method for "integers with the difference between their cubes is a square, and v.v.", Yahoo! answers, 2011
EXAMPLE
10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.
8954^3 - 5687^3 = 730719^2, 8954^2 - 5687^2 = 363^3.
PROG
(Python)
def cube(z, p):
iscube=False
y=int(pow(z, 1/p)+0.01)
if y**p==z:
iscube=True
return iscube
for n in range (1, 10**5):
for m in range(n+1, 10**5):
a=(m-n)*(m**2+m*n+n**2)
b=(m-n)*(m+n)
if cube(a, 2)==True and cube(b, 3)==True:
print (n, m)
CROSSREFS
KEYWORD
nonn
AUTHOR
Pieter Post, Aug 14 2015
EXTENSIONS
Added a(6) and more terms from Chai Wah Wu, Aug 17 2015
STATUS
approved