169 as Sum of up to 155 Squares/Mistake

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$169$

Mistake

In fact, $169$ can be written as the sum of $n$ non-zero squares, for all values of $n$ from $1$ to $155$, but for no larger values. [ Jackson, Masat and Mitchell, MM v61 41 ]

This is apparently untrue: $169$ can be expressed as the sum of $n$ non-zero squares for several values of $n$ greater than $155$, for example:

\(\ds 169\)

\(=\)

\(\ds 169 \times 1^2\)

$169$ squares

\(\ds 169\)

\(=\)

\(\ds 2^2 + 165 \times 1^2\)

that is, $166$ squares

\(\ds 169\)

\(=\)

\(\ds 2 \times 2^2 + 161 \times 1^2\)

that is, $163$ squares

\(\ds 169\)

\(=\)

\(\ds 3^2 + 160 \times 1^2\)

that is, $161$ squares

\(\ds 169\)

\(=\)

\(\ds 3^2 + 2^2 + 156 \times 1^2\)

that is, $158$ squares

The citation given is also wrong. It should refer to volume $66$, not $61$:

Feb. 1993: Kelly Jackson, Francis Masat and Robert Mitchell: Extensions of a Sums-of-Squares Problem (Math. Mag. Vol. 66, no. 1: pp. 41 – 43)  www.jstor.org/stable/2690474

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $169$