Solving

is a triangular number

,

Here are the first few solutions:

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There is an unusual pattern amongst these numbers, there always seems to be many more solutions when the integer length of x is 9, in other words 18 digit concatenated numbers. for all types of number i.e. square, triangular, pentagonal, octagonal… … up to hexagonal type numbers. and the hexagonal numbers are the first that do not have a 20 digit solution.

Paul.

Hexagonal numbers: n*(2*n-1)

when integer length of x is 9

(10^9 + 1)*x + 1 = y*(2*y-1)

when integer length of x is 10

(10^10 + 1)*x + 1 = y*(2*y-1)

lots of solutions

And (10^n + 1)*x, n ≥ 9 ( I have checked for n up to 20 and for Triangular,Pentagonal,Hexagonal,Heptagonal,Octagonal and Nonagonal

I should say (10^n + 1)*x + 1

Sorry I meant to say octagonal numbers, have no 20 digit solutions, (x=10). y(3y -2) type.

P.