Modulo Arithmetic/Examples/a^2 + (a+2)^2 + (a+4)^2 + 1 Modulo 12

Example of Modulo Arithmetic

Let $a$ be an odd integer.

Then:

$a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1 \equiv 0 \pmod {12}$

Let $a = 2 k + 1$ where $k \in \Z$.

Then:

$\ds a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1$

$=$

$\ds \paren {2 k + 1}^2 + \paren {2 k + 3}^2 + \paren {2 k + 5}^2 + 1$

$\ds $

$=$

$\ds 4 k^2 + 4 k + 1 + 4 k^2 + 12 k + 9 + 4 k^2 + 20 k + 25 + 1$

$\ds $

$=$

$\ds 12 k^2 + 36 k + 36$

$\ds $

$=$

$\ds 12 \paren {k^2 + 3 k + 3}$

Hence the result.

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-1}$ Euclid's Division Lemma: Exercise $7$