Introduction

A Diophantine equation is an equation in which only integer solutions are allowed.

3x+5y=1 has many solutions. One of the solution is: x₀=2, y₀=-1. Other solutions can be found by adding fives to x₀ and subtracting threes from y₀. (x,y)=[(2,-1), (7,-4), (12,-7), ..., (-3,2), ...]. 

4x+2y=1 has no integer solutions, since the left side is even for any choice of integers and the right side is always odd.

9x+12y=4 has no integer solutions, since the left side is divisible by 3 and the right side isn't.

In general, one might need a special computer code to solve a given Diophantine equation.

Triangle, and hexagonal numbers are generated by the following formulae:

• Triangle -  T_t=t(t+1)/2   1, 3, 6, 10, 15, ...
• Hexagonal - H_h=h(2h−1)    1, 6, 15, 28, 45, ...

We see that T₁=H₁,  T₃=H₂ and T₅=H₃.

When is an hexagonal number equal triangle number?

We start with assuming that t is even, so we write: t=2t₀.

Thus, we have to solve:
2t₀(2t₀+1)/2 = h(2h-1) => open brackets
2t₀²+t₀ = 2h²-h => 
h+t₀ = 2h² - 2t₀² = 2(h+t₀)*(h-t₀) => divide
1 = 2(h-t₀) =>
0.5 = h-t₀
this equation has NO integer solutions.
So, it must be odd.

Now assume, t = 2t₁-1. Thus, we have to solve:
(2t₁-1)*(2t₁)/2 = h*(2h-1) => open brackets
t₁*(2t₁-1) = h*(2h-1) => t₁ = h.

So, t = 2t₁-1 = 2h-1 => T_2h-1=H_h.

And this is the final answer.