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[ September 12, 2025 ] [ Last update – ]
Curious sums with fifthpowers.
By Patrick De Geest


Take a look at this pattern
\begin{aligned} 50^5-5^5&=312496875\\ 500^5-50^5-5^5&=31249687496875\\ 5000^5-500^5-50^5-5^5&=3124968749687496875\\ 50000^5-5000^5-500^5-50^5-5^5&=312496874968749687496875\\ 500000^5-50000^5-5000^5-500^5-50^5-5^5&=31249687496874968749687496875\\ \cdots&=\cdots \end{aligned} In the result is a reappearing part of \(5\) digits \(49687\)
wedged between \(312\) and \(5\). Together they form a ninedigital.
We can write a shorthand result using a vinculum \(312\,{\color{red}{\overline{ {\color{black}{49687}} }}}_{\color{red}{5}}\,5\)

Turning the terms all positive gives \begin{aligned} 50^5+5^5&=312503125\\ 500^5+50^5+5^5&=31250312503125\\ 5000^5+500^5+50^5+5^5&=3125031250312503125\\ 50000^5+5000^5+500^5+50^5+5^5&=312503125031250312503125\\ 500000^5+50000^5+5000^5+500^5+50^5+5^5&=31250312503125031250312503125\\ \cdots&=\cdots \end{aligned} In shorthand we write \(\sum_{n=0}^{k}\;(5*10^n)^5=\overline{31250}_{n}\,3125\)
Note that between each pair of \(3125\) we see a zero separator.

A000235 Prime Curios! Prime Puzzle
Wikipedia 235 Le nombre 235

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