Henry Ernest Dudeney/Modern Puzzles/23 - De Morgan and Another

by : $23$

 * De Morgan and Another
 * , the mathematician, who died in $1871$, used to boast that he was $x$ years old in the year $x^2$.
 * My living friend, Jasper Jenkins, wishing to improve on this, tells me he was $a^2 + b^2$ in $a^4 + b^4$;
 * that he was $2 m$ in the year $2 m^2$;
 * and that he was $3 n$ years old in the year $3 n^4$.


 * Can you give the years in which and Jenkins were respectively born?

Solution
was born in $1806$.

Jasper Jenkins was born in $1860$.

Proof
Note that was writing this in the $1920$s.

The square numbers around the $18$th and $19$th century are:


 * $42^2 = 1764$
 * $43^2 = 1849$
 * $44^2 = 1936$

of which only $1849 = 43^2$ can plausibly fit the parameters for.

Hence it is deduced that was born on $1849 - 43 = 1806$.

When we inspect his page, we see that indeed he was born on $18$th June $1806$.

As for Jasper, we need to inspect square numbers around the $900$ to $1000$ region:


 * $2 \times 30^2 = 2 \times 900 = 1800$
 * $2 \times 31^2 = 2 \times 961 = 1922$
 * $2 \times 32^2 = 2 \times 1024 = 2048$

Clearly Jasper was $2 \times 31 = 62$ in $1922$.

Hence it appears Jasper was born in $1860$.

We check the $4$th powers over the range $600$ to $700$ and find:


 * $3 \times 5^3 = 3 \times 625 = 1875$

which corroborates the above: Jasper was $3 \times 5 = 15$ in $1875$.

Continuing to explore the $4$th powers, we have this list:
 * $1^4 = 1$
 * $2^4 = 16$
 * $3^4 = 81$
 * $4^4 = 256$
 * $5^4 = 625$
 * $6^4 = 1296$
 * $7^4 = 2401$

and we have gone high enough.

Inspecting these numbers, we have that:
 * $625 + 1296 = 1921$

at which time Jasper was $5^2 + 6^2 = 25 + 36 = 61$.