Divisibility of Sum of 3 Fourth Powers
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Theorem
Let $n \in \Z_{\ge 0}$ be the sum of three $4$th powers.
Then:
- $n$ is divisible by $5$ if and only if all three addends are also divisible by $5$
- $n$ is divisible by $29$ if and only if all three addends are also divisible by $29$.
Proof
Let $n = a^4 + b^4 + c^4$ for $a, b, c \in \Z$.
Necessary Condition
Let $a$, $b$ and $c$ all be divisible by either $5$ or $29$.
That is:
- $a, b, c \equiv 0 \pmod 5$
or:
- $a, b, c \equiv 0 \pmod {29}$
We have that for integer $p$:
\(\ds a, b, c\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod p\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^4, b^4, c^4\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod p\) | Congruence of Powers | |||||||||
\(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod p\) | Modulo Addition is Well-Defined | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds p\) | \(\divides\) | \(\ds a^4 + b^4 + c^4\) | where $\divides$ denotes divisibility |
Hence if all three addends are divisible by either $5$ or $29$, then so is the sum.
$\Box$
Sufficient Condition
Let $n$ be divisible by $5$.
Then:
- $a^4 + b^4 + c^4 \equiv 0 \mod 5$
From Fourth Power Modulo 5:
- $x \in \Z_5 \implies x^4 \in \set {0, 1}$
So:
- $a^4 + b^4 + c^4 \equiv 0 \mod 5 \implies a, b, c \equiv 0 \mod 5$
$\Box$
Let $n$ be divisible by $29$.
Then $a^4 + b^4 + c^4 \equiv 0 \mod {29}$.
But for $x \in \Z_{29}$, $x^4 \in \set {0, 1, 7, 16, 20, 23, 24, 25} = R$ necessarily.
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So:
- $a^4 + b^4 + c^4 \equiv 0 \mod {29} \implies a, b, c \equiv 0 \mod {29}$
as $29$ and its non-zero multiples are not partitioned by any 3 elements in $R$
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$\blacksquare$
Historical Note
This result was established by Leonhard Paul Euler.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $29$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $29$