Sum of Two Cubes

Summary: The proposition that $t^3+x^3=z^3$ has no nontrivial integer solutions — i.e., the sum of two positive integer cubes is never a cube. Cited by Euler as established (footnote 81) when explaining why $1+x^3$ admits no further cube values beyond $x=0,-1$. This is Fermat’s Last Theorem at exponent $n=3$; Euler proves it in detail in ch2.0.15-questions-cubes §243 — see fermats-last-theorem-n3.

Sources: chapter-2.0.10, chapter-2.0.15

Last updated: 2026-05-09

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Statement

For positive integers $t$, $x$, $z$:

$t^3+x^3\ne z^3.$

Equivalently, $1+x^3$ (taking $t=1$) is never a perfect cube for $x\ge 1$, and the only integer solutions of ”$1+x^3$ is a cube” are $x=0$ (giving $1$) and $x=-1$ (giving $0$).

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Use in Elements of Algebra

Euler invokes the result in ch2.0.10-cubic-formula-as-cube §155 to confirm that the bootstrap method exhausts the solutions of ”$1+x^3$ is a cube” at $\{0,-1\}$. The bootstrap from $x=0$ recovers only $x=-1$, and substituting $x=-1+y$ produces a Case-2 formula whose ansatz forces $y=\infty$, leaving no further integer values. The general impossibility theorem then closes the question.

By the same logic, $x^3+2$ is a cube only at $x=-1$ (§156), since otherwise one would have $1^3+2=(\text{cube}{)}^3-(-1{)}^3$, again contradicting the sum-of-two-cubes theorem after rearrangement.

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Historical Context

• Fermat (1637) asserted in the margin of his Diophantus that $a^n+b^n=c^n$ has no nontrivial integer solutions for $n>2$, but published no proof.
• Euler (1770), the same year as Elements of Algebra, gave the first widely-accepted proof for $n=3$ in his Vollständige Anleitung zur Algebra (the very treatise these wiki pages summarize — see ch2.0.15-questions-cubes §243 and fermats-last-theorem-n3 for the proof). Uses infinite descent and arithmetic in $\mathbb{Z}[\sqrt{-3}]$ (with a now-known gap involving unique factorization in that ring; the gap is repaired by working in $\mathbb{Z}[\omega ]$).
• The general theorem (“Fermat’s Last Theorem”) was finally proved by Andrew Wiles in 1995, two centuries after Euler.

The footnote in §155 (“it may be demonstrated that the sum of two cubes can never become a cube”) refers to this theorem as known, consistent with Euler’s own 1770 publication.

Companion Theorems (§247)

In the same chapter Euler proves the parallel statement that

$x^3\pm y^3\ne 2z^3,$

i.e., neither sum nor difference of two cubes can equal twice a cube (except trivially when $y=x$). The proof reuses the descent skeleton of §243.

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Connection to Surd Rationalization

The sum-of-two-cubes impossibility explains a structural failure of the bootstrap technique in ch2.0.10-cubic-formula-as-cube:

• For most cubic-radicand-as-cube problems, a single seed solution generates an infinite cascade via the Case-1/2/3 ansätze.
• For $t^3+x^3$, the impossibility theorem forces the cascade to terminate after exhausting the trivial roots, regardless of which seed is chosen.

This is the cube-analog of impossibility phenomena treated in ch2.0.5-impossibility-quadratic-squares for $a{t}^2+b{u}^2$ via residue-class analysis.

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Related pages

• ch2.0.10-cubic-formula-as-cube
• ch2.0.15-questions-cubes
• ch2.0.5-impossibility-quadratic-squares
• ch2.0.13-impossibility-biquadrate-sums
• fermats-last-theorem-n3
• fermats-last-theorem-n4
• three-cubes-as-cube
• infinite-descent
• cubic-equations
• cube-roots
• indeterminate-analysis