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 Canadian mathematician [[Irving Kaplansky]] (1917-2006).<ref>See: I. Kaplansky, ''The forms ''x''+32''y''² and ''x''+64''y''²'' [sic],{{citation

Procedings of the American Mathematical Society '''131''' (2003), no. 7, 2299--2300.</ref>

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

| doi = 10.1090/S0002-9939-03-07022-9

| issue = 7

| journal = [[Proceedings of the American Mathematical Society]]

| mr = 1963780

| pages = 2299–2300 (electronic)

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

| volume = 131

| year = 2003| doi-access = free

}}.</ref>

==Statement of the theorem==

Kaplansky's theorem states that a prime ''p'' [[Modular arithmetic|congruent to 1 modulo 16]] is representable by both or none of ''x''²&nbsp;+&nbsp;32''y''² and ''x''²&nbsp;+&nbsp;64''y''², whereas a prime ''p'' congruent to 9 modulo 16 is representable by exactly one of these quadratic forms.

This is remarkable since the primes represented by each of these forms individually are ''not'' describable by congruence conditions.<ref>{{citation

Kaplansky's theorem states that a prime ''p'' [[Modular arithmetic|congruent to 1 modulo 16]] is representable by both or none of ''x''²+32''y''² and ''x''²+64''y''², whereas a prime ''p'' congruent to 9 modulo 16 is representable by exactly one of these quadratic forms.

| last = Cox | first = David A.

| isbn = 0-471-50654-0

This is remarkable since the primes represented by each of these forms individually are

| ___location = New York

''not'' describable by congruence conditions.<ref>See: D. A. Cox, ''Primes of the Form ''x''²+''ny''²'', Wiley, New York, 1989.</ref>

| mr = 1028322

| publisher = John Wiley & Sons

| title = Primes of the form {{math|x² + ny²}}

| year = 1989}}.</ref>

==Proof==

Kaplansky's proof uses the facts that 2 is a 4th power modulo ''p'' if and only if ''p'' is representable by ''x''²&nbsp;+&nbsp;64''y''², and that -&minus;4 is an 8th power modulo &nbsp;''p'' if and only if ''p'' is representable by ''x''²&nbsp;+&nbsp;32''y''².

Kaplansky's proof uses the facts that 2 is a 4th power modulo ''p'' if and only if ''p'' is representable by ''x''²+64''y''², and that -4 is an 8th power modulo ''p'' if and only if ''p'' is representable by ''x''²+32''y''².

==Examples==

*The prime ''p''&nbsp;=41&nbsp;17 is congruent to 91 modulo 16 and is representable by neither ''x''²&nbsp;+&nbsp;32''y''² (since 41=3²+32&times;1²), but not bynor ''x''²&nbsp;+&nbsp;64''y''².

*The prime ''p''&nbsp;=17&nbsp;113 is congruent to 1 modulo 16 and is representable by neitherboth ''x''²&nbsp;+&nbsp;32''y''² norand ''x''²+64''y''² (since 113&nbsp;=&nbsp;9²&nbsp;+&nbsp;32×1² and 113&nbsp;=&nbsp;7²&nbsp;+&nbsp;64×1²).

*The prime ''p''&nbsp;=73&nbsp;41 is congruent to 9 modulo 16 and is representable by ''x''²&nbsp;+64&nbsp;32''y''² (since 7341&nbsp;=&nbsp;3²&nbsp;+64&timesnbsp;132×1²), but not by ''x''²&nbsp;+32&nbsp;64''y''².

*The prime ''p''&nbsp;=113&nbsp;73 is congruent to 19 modulo 16 and is representable by both ''x''²+32''y''² and ''x''²&nbsp;+&nbsp;64''y''² (since 11373&nbsp;=9&nbsp;3²&nbsp;+32&timesnbsp;164×1²), andbut 113=7not by ''x''²&nbsp;+64&timesnbsp;132''y''²).

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

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

==Similar results==

Five results similar to Kaplansky's theorem are known:<ref>{{citation

| last = Brink | first = David

| doi = 10.1016/j.jnt.2008.04.007

| issue = 2

| journal = [[Journal of Number Theory]]

| mr = 2473893

| pages = 464–468

Five results similar to Kaplansky's theorem are known<ref>See: | D.title Brink,= ''Five peculiar theorems on simultaneous representation of primes by quadratic forms'',

| volume = 129

| year = 2009| doi-access =

}}.</ref>

*A prime ''p'' congruent to 1 or 169 modulo 24020 is representable by both or none of ''x''²&nbsp;+150&nbsp;20''y''² and ''x''²&nbsp;+960&nbsp;100''y''², whereas a prime ''p'' congruent to 49 or 1219 modulo 24020 is representable by exactly one of these quadratic forms.

Five results similar to Kaplansky's theorem are known<ref>See: D. Brink, ''Five peculiar theorems on simultaneous representation of primes by quadratic forms'',

*A prime ''p'' congruent to 1, 16 or 22 modulo 39 is representable by both or none of ''x''²&nbsp;+&nbsp;''xy''&nbsp;+&nbsp;10''y''² and ''x''²&nbsp;+&nbsp;''xy''&nbsp;+&nbsp;127''y''², whereas a prime ''p'' congruent to 4, 10 or 25 modulo 39 is representable by exactly one of these quadratic forms.

Journal of Number Theory '''129''' (2009), 464-468.</ref>:

*A prime ''p'' congruent to 1, 16, 26, 31 or 36 modulo 55 is representable by both or none of ''x''²&nbsp;+&nbsp;''xy''&nbsp;+&nbsp;14''y''² and ''x''²&nbsp;+&nbsp;''xy''&nbsp;+&nbsp;69''y''², whereas a prime ''p'' congruent to 4, 9, 14, 34 or 49 modulo 55 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1, 65 or 81 modulo 20112 is representable by both or none of ''x''²&nbsp;+20&nbsp;14''y''² and ''x''²&nbsp;+100&nbsp;448''y''², whereas a prime ''p'' congruent to 9, 25 or 57 modulo 20112 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1, 65 or 81169 modulo 112240 is representable by both or none of ''x''²&nbsp;+14&nbsp;150''y''² and ''x''²&nbsp;+448&nbsp;960''y''², whereas a prime ''p'' congruent to 9, 2549 or 57121 modulo 112240 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1, 16 or 22 modulo 39 is representable by both or none of ''x''²+''xy''+10''y''² and ''x''²+''xy''+127''y''², whereas a prime ''p'' congruent to 4, 10 or 25 modulo 39 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1, 16, 26, 31 or 36 modulo 55 is representable by both or none of ''x''²+''xy''+14''y''² and ''x''²+''xy''+69''y''², whereas a prime ''p'' congruent to 4, 9, 14, 34 or 49 modulo 55 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1, 65 or 81 modulo 112 is representable by both or none of ''x''²+14''y''² and ''x''²+448''y''², whereas a prime ''p'' congruent to 9, 25 or 57 modulo 112 is representable by exactly one of these quadratic forms.

*A prime ''p'' congruent to 1 or 169 modulo 240 is representable by both or none of ''x''²+150''y''² and ''x''²+960''y''², whereas a prime ''p'' congruent to 49 or 121 modulo 240 is representable by exactly one of these quadratic forms.

It is conjectured that there are no other similar results involving definite forms.

==Notes==

{{reflist}}

[[Category:MathematicalTheorems theoremsin number theory]]

[[Category:Quadratic forms]]