Kaplansky's theorem on quadratic forms: Difference between revisions

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{{short description|Result on simultaneous representation of primes by quadratic forms}}

In [[mathematics]], '''Kaplansky's theorem on quadratic forms''' is a result on simultaneous representation of [[Prime number|primes]] by [[quadratic forms]]. It was proved in 2003 by [[Irving Kaplansky]].<ref>{{citation

| last = Kaplansky | first = Irving | authorlink = Irving Kaplansky

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| title = The forms {{math|''x'' + 32''y''² and ''x'' + 64''y''^2}} {{sic}}

| volume = 131

| year = 2003}}.</ref>| doi-access = free

}}.</ref>

==Statement of the theorem==

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==Examples==

*The prime ''p''&nbsp;=&nbsp;17 is congruent to 1 modulo 16 and is representable by neither ''x''²&nbsp;+&nbsp;32''y''² nor ''x''²&nbsp;+&nbsp;64''y''².

*The prime ''p''&nbsp;=&nbsp;113 is congruent to 1 modulo 16 and is representable by both ''x''²&nbsp;+&nbsp;32''y''² and ''x''²+64''y''² (since 113&nbsp;=&nbsp;9²&nbsp;+&nbsp;32&times;132×1² and 113&nbsp;=&nbsp;7²&nbsp;+&nbsp;64&times;164×1²).

*The prime ''p''&nbsp;=&nbsp;41 is congruent to 9 modulo 16 and is representable by ''x''²&nbsp;+&nbsp;32''y''² (since 41&nbsp;=&nbsp;3²&nbsp;+&nbsp;32&times;132×1²), but not by ''x''²&nbsp;+&nbsp;64''y''².

*The prime ''p''&nbsp;=&nbsp;73 is congruent to 9 modulo 16 and is representable by ''x''²&nbsp;+&nbsp;64''y''² (since 73&nbsp;=&nbsp;3²&nbsp;+&nbsp;64&times;164×1²), but not by ''x''²&nbsp;+&nbsp;32''y''².

==Similar results==

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| title = Five peculiar theorems on simultaneous representation of primes by quadratic forms

| volume = 129

| year = 2009}}.</ref>| doi-access =

}}.</ref>

*A prime ''p'' congruent to 1 modulo 20 is representable by both or none of ''x''²&nbsp;+&nbsp;20''y''² and ''x''²&nbsp;+&nbsp;100''y''², whereas a prime ''p'' congruent to 9 modulo 20 is representable by exactly one of these quadratic forms.