Kaplansky's theorem on quadratic forms
theorem that a prime congruent to 1 modulo 16 is representable by either both or neither of the quadratic forms x²+32y² and x²+64y², while a prime congruent to 9 modulo 16 is representable by exactly one of the two
| rdfs:label |
"Kaplansky's theorem on quadratic forms"@en
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| schema:description |
"theorem that a prime congruent to 1 modulo 16 is representable by either both or neither of the quadratic forms x²+32y² and x²+64y², while a prime congruent to 9 modulo 16 is representable by exactly one of the two"@en
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schema:date |
"2025-08-29T16:09:07+00:00"^^xsd:dateTime |
| schema:version | 2398130971 |
| wikibase:identifiers | 1 |
| wikibase:sitelinks | 1 |
| wikibase:statements | 6 |
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| p:P2534 | wd:statement/Q17098379-d1a7062b-4a17-b64d-4be0-c41e9fa73f83 |
| p:P31 | wd:statement/Q17098379-757BD165-D9B3-4927-AFC0-E887C5615014 |
| p:P361 | wd:statement/Q17098379-C34B7D1F-98E5-43E2-AD79-456FCFC8BA85 |
| p:P6104 | wd:statement/Q17098379-6EF62723-385B-4C3C-B00B-A811989560C7 |
| p:P646 | wd:statement/Q17098379-32AC3EDF-3D81-459D-ACCE-E0EE74D6F958 |
| wdt:P1318 | wd:Q129368 |
| wdt:P2534 | "<math class="mwe-math-element mwe-math-element-inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd><mi>p</mi><mo stretchy="false">≡</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mrow data-mjx-texclass="ORD"><mn>1</mn><mn>6</mn></mrow><mo stretchy="false">)</mo></mrow></mtd><mtd><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo>⟹</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">∈</mo><mrow data-mjx-texclass="ORD"><mi>ℤ</mi></mrow><mo stretchy="false">:</mo><mi>p</mi><mo stretchy="false">=</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mn>3</mn><mn>2</mn><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo>⟺</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">∈</mo><mrow data-mjx-texclass="ORD"><mi>ℤ</mi></mrow><mo stretchy="false">:</mo><mi>p</mi><mo stretchy="false">=</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mn>6</mn><mn>4</mn><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd><mi>p</mi><mo stretchy="false">≡</mo><mn>9</mn><mrow data-mjx-texclass="ORD"><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mrow data-mjx-texclass="ORD"><mn>1</mn><mn>6</mn></mrow><mo stretchy="false">)</mo></mrow></mtd><mtd><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo>⟹</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">∈</mo><mrow data-mjx-texclass="ORD"><mi>ℤ</mi></mrow><mo stretchy="false">:</mo><mi>p</mi><mo stretchy="false">=</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mn>3</mn><mn>2</mn><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo>⟺</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>∄</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">∈</mo><mrow data-mjx-texclass="ORD"><mi>ℤ</mi></mrow><mo stretchy="false">:</mo><mi>p</mi><mo stretchy="false">=</mo><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mn>6</mn><mn>4</mn><msup><mi>y</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr></mtable></mrow></mstyle></mrow></math>"^^http://www.w3.org/1998/Math/MathML |
| wdt:P31 | wd:Q65943 |
| wdt:P361 | wd:Q944443 |
| wdt:P6104 | wd:Q8487137 |
| wdt:P646 | "/m/05b1dwb" |
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