Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239/Mistake

Source Work

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

The Dictionary

$239$

Mistake

The 'approximation' to a Fermat equation, $x^4 + y^4 = z^4 + 1$, has $3$ solutions with $x = 239$. The other numbers are $y = 104, z = 58, 136$; $y = 143, z = 60,671$; $y = 208, z = 71, 656$.

Correction

There are $2$ points here:

$(1): \quad$ The equation in question is $x^4 + y^4 = z^2 + 1$.

$(2): \quad$ It may not necessarily be the case that there are only $3$ solutions. The cited article claims only that these are the only $3$ solutions where $x > y$.

Sources

• 1986: Joseph McLean: On Sums of Unlike Powers (Mathematical Spectrum Vol. 18, no. 3: pp. 88 – 89)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $239$