How to find the 4th special number among pentagonal and triangular numbers

A pentagonal number is defined by p(n) = (3n^2 – n)/2, where n is an integer starting from 1. Therefore, the first 12 pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176 and 210.

A triangular number is defined by t(n) = (n^2 + n)/2, where n is an integer starting from 1. Therefore, the first 12 triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 and 78.

From the above example, the number 1 is considered 'special' because it is both a pentagonal (p=1) and a triangular number (t=1). Determine the 4th 'special' number (i.e. where p=t) that exists assuming the number 1 to the be first 'special' number.

Answer 1

[Chunru]

n = 1:5000;
p = (3*n.^2 - n) / 2;
t = (n.^2 + n ) /2;
a = intersect(p, t);
disp(a)

The special number is 7906276