Infinite descent

Infinite descent is a brilliant method of mathematical proof promoted by Pierre de Fermat in the 1600s. This approach facilitates elegant proofs of challenging problems, and can be tremendously useful beyond mathematics too.

This approach is analogous to proof by induction except infinite descent goes in the backwards, or downwards, direction until a contradiction occurs, thereby proving the falsehood of the hypothesis. The approach can be summed up as this:

Assume that a solution to an equation exists among positive integers. Then show that the existence of that solution implies the existence of another solution with smaller natural positive integers. Then show that yet another solution among positive integers must exist, and so on ad infinitum. But descent ad infinitum is impossible among positive integers, and hence a contradiction occurs. Therefore no solution can exist among positive integers.

A better name for "infinite descent" as used in mathematics could be "descent until contradiction," because the point of infinite descent is to show that if a hypothesis were true, then an impossible infinite consent would be possible. The contradiction disproves the hypothesis.

Infinite descent and the Bible

The insights from an approach of backward descent is hinted at in logic referenced by Jesus:

“

Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.”

”

Jesus did not say, "I was," as one might expect, but rather "I am." In other words, unlimited infinite descent is available to Jesus, just as multiplication of the loaves is.

Theorems proven by infinite descent

Leonard Euler proved by infinite descent Fermat's unsolved theorem on the sums of two squares. That theorem states that an odd prime number p is the sum of the following squares of integers x and y, such that p = x² + y², if and only if p is congruent to 1 (mod 4).

See also

• infinity