Numbers whose Difference equals Difference between Cube and Seventh Power
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Theorem
The following $2$ pairs of integers are the only ones known which exhibit this pattern:
| \(\ds \size {5^3 - 2^7}\) | \(=\) | \(\ds 5 - 2\) | ||||||||||||
| \(\ds \size {13^3 - 3^7}\) | \(=\) | \(\ds 13 - 3\) |
Proof
We have:
| \(\ds \size {5^3 - 2^7}\) | \(=\) | \(\ds \size {125 - 128}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5 - 2\) |
| \(\ds \size {13^3 - 3^7}\) | \(=\) | \(\ds \size {2197 - 2187}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 - 3\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $125$