Sum of 4 Consecutive Binomial Coefficients forming Square/Mistake

Source Work

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

The Dictionary

$767$

Mistake

Mistake $1$:

$\dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3 + \dbinom {767} 4$ is a perfect square, $8672^2$.

Mistake $2$:

The smaller solutions are $7$, $15$ and $74$.

Correction

The first mistake is that it is $\dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3 = 8672^2$.

The second mistake is that the numbers $-1$, $0$ and $2$ have the same property.

Wells is perhaps excused the first two, as they may not have been in his purview, but omitting $2$ is a less forgivable oversight, considering the complete set is given in 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.), which he himself references.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $767$