Pentagonal Triangular Number
A number which is simultaneously a pentagonal number 
and triangular number 
. Such numbers exist when
 |
(1)
|
Completing the square gives
 |
(2)
|
Substituting 
and 
gives the Pell-like quadratic Diophantine equation
 |
(3)
|
which has solutions 
, (19, 11), (71, 41), (265, 153), .... In terms of 
, these give (1, 1), (10/3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the whole number solutions are 
, (12, 20), (165, 285), (2296, 3976), ... (OEIS A046174 and A046175), corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805, ... (OEIS A014979).
REFERENCES:
Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996.
Sloane, N. J. A. Sequences A014979, A046174, and A046175 in "The On-Line Encyclopedia of Integer Sequences."
