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&ndash;2006).<ref>See: I. Kaplansky, ''The forms x&nbsp;+&nbsp;''32''y^''2'' and x&nbsp;+&nbsp;''64''y² [sic],{{citation

Proceedings of the American Mathematical Society '''131''' (2003), no. 7, 2299&ndash;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 '''131''' (2003), no. 7, 2299&ndash;2300.</ref>]]

| 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>See: D. A. Cox, ''Primes of the Form ''x''²&nbsp;+&nbsp;''ny''²'', Wiley, New York, 1989.</ref>{{citation

| last = Cox | first = David A.

| isbn = 0-471-50654-0

| ___location = New York

| 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.

==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;41113 is congruent to 91 modulo 16 and is representable by both ''x''²&nbsp;+&nbsp;32''y''² and ''x''²+64''y''² (since 41113&nbsp;=&nbsp;39²&nbsp;+&nbsp;32&times;132×1²), butand not by ''x''113&nbsp;=&nbsp;7²&nbsp;+&nbsp;64''y''64×1²).

*The prime ''p''&nbsp;=113&nbsp;41 is congruent to 19 modulo 16 and is representable by both ''x''²&nbsp;+&nbsp;32''y''² and ''x''²+64''y''² (since 11341&nbsp;=&nbsp;93²&nbsp;+&nbsp;32&times;132×1²), andbut not by 113&nbsp;=&nbsp;7''x''²&nbsp;+&nbsp;64&times;1''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.

*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;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;1²), but not by ''x''²&nbsp;+&nbsp;32''y''².

==Similar results==

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

| last = Brink | first = David

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

| doi = 10.1016/j.jnt.2008.04.007

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

| 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 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.

*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.

*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 112 is representable by both or none of ''x''²&nbsp;+&nbsp;14''y''² and ''x''²&nbsp;+&nbsp;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&nbsp;+&nbsp;150''y''² and ''x''²&nbsp;+&nbsp;960''y''², whereas a prime ''p'' congruent to 49 or 121 modulo 240 is representable by exactly one of these quadratic forms.

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==Notes==

{{reflist}}