Sum of Square Numbers (四平方数和定理)

Sum of Square Numbers (四平方数和定理)

日付: 7月 18, 2019

An arbitrary number <span><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>∈</mo><msup><mi>N</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">n \in N^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5782em; vertical-align: -0.0391em;"></span><span class="mord mathit">n</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.771331em; vertical-align: 0em;"></span><span class="mord"><span class="mord mathit" style="margin-right: 0.10903em;">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.771331em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span> can be represented as the sum of at most 4 square numbers.
If <span><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>=</mo><msup><mn>4</mn><mi>a</mi></msup><mo>(</mo><mn>8</mn><mi>k</mi><mo>+</mo><mn>7</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">n = 4^a(8k + 7)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">n</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.664392em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">a</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">8</span><span class="mord mathit" style="margin-right: 0.03148em;">k</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">7</span><span class="mclose">)</span></span></span></span></span>, <span><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathit">n</span></span></span></span></span> can only be splitted into 4 square numbers.

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