Equations in Integers

An equation in integers (or Diophantine equation) is one where we require solutions to be whole numbers — usually integers or natural numbers.

The key difference from ordinary algebra: we can no longer divide freely or take square roots. Instead, the structure of the integers themselves — divisibility, parity, and modular arithmetic — becomes the main tool.

Core strategy: factor and list divisor pairs

Many integer equations can be rearranged into the form

$AB = N$

where $A$ and $B$ are integer expressions and $N$ is a known constant. Since $N$ has finitely many divisor pairs, we list them all and check which ones give valid solutions.

Example. Solve $x^2 - y^2 = 91$ in natural numbers.

Rewrite as $(x-y)(x+y) = 91$. The positive divisor pairs of $91$ are $(1, 91)$ and $(7, 13)$. Setting $x - y = 1$, $x + y = 91$ gives $x = 46$, $y = 45$. Setting $x - y = 7$, $x + y = 13$ gives $x = 10$, $y = 3$.

Parity arguments

When both sides of an equation must match in parity (odd/even), many candidate solutions are eliminated.

Example. If $x$, $y$, $z$ are all odd, then $x + y$ and $x + z$ are even while $y + z$ is also even. Checking whether $(x+y)^2 + (x+z)^2 = (y+z)^2$ is possible modulo $4$ often reveals a contradiction.

Modular obstructions

Reduce the equation modulo a small number ($2$, $3$, $4$, $8$, $11$, ...) to show that no solution can exist, or to narrow the possibilities.

Example. Perfect squares are $\equiv 0$ or $1 \pmod{4}$. If an equation forces a square to be $\equiv 2 \pmod{4}$, there is no solution.

Problems in this set

The first three problems are for discussion — they introduce the factoring and parity techniques. The remaining problems are for independent work, ordered roughly by difficulty.