Pierādīt, ka nevienai naturālai \(n\) vērtībai izteiksmes
\[4^{n}+5^{n}+6^{n}+7^{n}+8^{n}+9^{n}+10^{n}+11^{n}+12^{n}+13^{n}\]
vērtība nav naturāla skaitļa kvadrāts!
Doto summu apzīmējam ar \(S\). Aplūkojam katru saskaitāmo un summu \(S\) pēc moduļa \(8\) dažādām \(n\) vērtībām.
| \(\boldsymbol{n}\) | \(\mathbf{1}\) | \(\mathbf{2}\) | \(\mathbf{3}\) | \(\mathbf{4}\) | \(\cdots\) |
|---|---|---|---|---|---|
| \(\mathbf{4}^{\boldsymbol{n}}\) | \(4\) | \(0\) | \(0\) | \(0\) | \(\cdots\) |
| \(\mathbf{5}^{\boldsymbol{n}}\) | \(5\) | \(1\) | \(5\) | \(1\) | \(\cdots\) |
| \(\mathbf{6}^{\boldsymbol{n}}\) | \(6\) | \(4\) | \(0\) | \(0\) | \(\cdots\) |
| \(\mathbf{7}^{\boldsymbol{n}}\) | \(7\) | \(1\) | \(7\) | \(1\) | \(\cdots\) |
| \(\mathbf{8}^{\boldsymbol{n}}\) | \(0\) | \(0\) | \(0\) | \(0\) | \(\cdots\) |
| \(\mathbf{9}^{\boldsymbol{n}}\) | \(1\) | \(1\) | \(1\) | \(1\) | \(\cdots\) |
| \(\mathbf{10}^{\boldsymbol{n}}\) | \(2\) | \(4\) | \(0\) | \(0\) | \(\cdots\) |
| \(\mathbf{11}^{\boldsymbol{n}}\) | \(3\) | \(1\) | \(3\) | \(1\) | \(\cdots\) |
| \(\mathbf{12}^{\boldsymbol{n}}\) | \(4\) | \(0\) | \(0\) | \(0\) | \(\cdots\) |
| \(\mathbf{13}^{\boldsymbol{n}}\) | \(5\) | \(1\) | \(5\) | \(1\) | \(\cdots\) |
| \(\boldsymbol{S}\) | \(\mathbf{5}\) | \(\mathbf{5}\) | \(\mathbf{5}\) | \(\mathbf{5}\) | \(\cdots\) |
Ievērojam, ka
Tātad visām \(n\) vērtībām summas \(S\) vērtība pēc moduļa \(8\) ir vienāda ar \(5\).
Naturālu skaitļu kvadrātu vērtības pēc moduļa \(8\) var būt tikai \(0,\ 1\) vai \(4\):
| \(n(\bmod 8)\) | \(n^{2}(\bmod 8)\) |
|---|---|
| \(0\) | \(0\) |
| \(1\) | \(1\) |
| \(2\) | \(4\) |
| \(3\) | \(1\) |
| \(4\) | \(0\) |
| \(5\) | \(1\) |
| \(6\) | \(4\) |
| \(7\) | \(1\) |
Tātad dotā izteiksme nevar būt naturāla skaitļa kvadrāts.