Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)

Theorem
The Diophantine equation
 * $n = x \paren {x + 1} = y \paren {y + 5} \paren {y + 10} \paren {y + 15}$

has exactly $2$ solutions in $\N \setminus \set 0$:

Proof
First, we observe that:

We note here that the square root of the product $x \paren {x + 1}$ has a fractional part which is less than one half.

Therefore, the fractional part of the square root of the product of the four integers on the must also be less than one half.

Next, we observe that:

Let $a = \paren {y^2 + 15 y + 25}$

We know that $\sqrt {a^2 - 25^2 } \lt a$

We can narrow the search for solutions tremendously as follows:

We need $\sqrt {a^2 - 25^2 } \lt a - \dfrac 1 2$ so that the fractional part of $\sqrt {a^2 - 25^2 }$ might be less than one half.

Therefore:

Therefore:

We now only need to review $18$ cases which are shown below.


 * [[File:Diophantine.png]]