Infinite Descent

Summary: A proof technique due to Fermat. To show that an equation has no positive-integer solution, assume one exists and construct a strictly smaller positive-integer solution from it. Iteration produces an infinite strictly decreasing sequence of positive integers, contradicting well-ordering. Used by Euler in ch2.0.13-impossibility-biquadrate-sums to prove $x^4\pm y^4\ne \square$, and by extension fermats-last-theorem-n4.

Sources: chapter-2.0.13, chapter-2.0.14, chapter-2.0.15

Last updated: 2026-05-09

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The Logical Skeleton

To prove a Diophantine equation $E(x_1,\dots ,x_k)=0$ has no solution in positive integers:

1. Assume a solution $(a_1,\dots ,a_k)$ exists.
2. Construct from it a new solution $(b_1,\dots ,b_k)$ with $0<\max (b_i)<\max (a_i)$.
3. Iterate: get an infinite sequence of solutions with strictly decreasing maxima.
4. Contradiction: positive integers cannot decrease forever (well-ordering of $\mathbb{N}$).
5. Conclude: no solution exists.

Equivalently (Fermat’s preferred phrasing): a “minimal counterexample” cannot exist, because the construction would yield a smaller one.

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Worked Application: $x^4+y^4=\square$

From ch2.0.13-impossibility-biquadrate-sums §205:

Stage	Variables	Relation
Start	$x,y,z$ coprime, $x$ odd, $y$ even	$x^4+y^4=z^2$
Pythagoras	$p,q$ coprime, $p$ odd, $q$ even	$x^2=p^2-q^2$, $y^2=2pq$
Pythagoras again	$r,s$ coprime	$p=r^2+s^2$, $q=2rs$, $x=r^2-s^2$
Coprime factors are squares	$t,u,v$	$r=t^2$, $s=u^2$, $r^2+s^2=v^2$
New equation		$t^4+u^4=v^2$

Since $t,u$ are derived from $x,y$ through fourth-root-like reductions, $\max (t,u)<\max (x,y)$. Descent.

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Why It’s Powerful

• No need to find solutions: the method proves non-existence without ever exhibiting a witness.
• Self-similarity: works best when the equation has a structure the descent step preserves (here: same shape $X^4+Y^4=Z^2$).
• Constructive: the descent step is explicit, not just an existence argument.

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Other Uses by Euler in this Wiki

• $x^4-y^4\ne \square$: same chapter, §206–208.
• $x^4+2y^4\ne \square$: same chapter, §210.
• $2x^4\pm 2y^4\ne \square$, $x^4\pm 4y^4\ne \square$, $4x^4-y^4\ne \square$: derived from the above by reduction.
• $p^2+q^2$ and $p^2+3q^2$ never both squares (residue-class pruning + descent): ch2.0.14-questions-squares §229-230.
• $x^3+y^3\ne z^3$ (FLT $n=3$): fermats-last-theorem-n3 / ch2.0.15-questions-cubes §243. Descent in $\mathbb{Z}[\sqrt{-3}]$ via $(t+u\sqrt{-3}{)}^3$ representation.
• $x^3\pm y^3\ne 2z^3$: ch2.0.15-questions-cubes §247. Same descent skeleton as §243.

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Where Descent Fails

Descent shows $x^4-2y^4\ne \square$ if applied naively — but in fact $x^4-2y^4=\square$ has infinitely many solutions ($3^4-2\cdot 2^4=49$, etc.). The reason: the auxiliary equation $r^2-2s^2=1$ (a Pell equation, pell-equation) has infinitely many solutions, so the “descent” loops back through Pell composition rather than terminating. See ch2.0.13-impossibility-biquadrate-sums §211.

This is a recurring caveat: descent works only when the auxiliary equations of each step are more restrictive, not equally permissive.

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Historical

• Pierre de Fermat (c. 1640) named the method descente infinie and used it for the $n=4$ FLT case among others. He claimed many results by this method without writing proofs.
• Euler systematized and published descent proofs, including the FLT $n=3$ and $n=4$ cases.
• Modern descent in arithmetic geometry — used in Mordell-Weil and elliptic curves — generalizes Fermat’s idea to algebraic groups.

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Related pages

• fermats-last-theorem-n3
• fermats-last-theorem-n4
• ch2.0.13-impossibility-biquadrate-sums
• ch2.0.14-questions-squares
• ch2.0.15-questions-cubes
• sum-of-two-cubes
• ch2.0.5-impossibility-quadratic-squares
• indeterminate-analysis