Kaplansky's theorem on quadratic forms: Difference between revisions

Revision as of 21:43, 28 April 2012

Latest revision as of 23:16, 18 August 2023

(11 intermediate revisions by 9 users not shown)

Line 1:

{{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>{{citation

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

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

Line 8 ⟶ 9:

| 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==

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.

Line 20 ⟶ 21:

| mr = 1028322

| publisher = John Wiley & Sons

| title = Primes of the form {{math|''x''² + ''ny''²}}

| 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

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

| last = Brink | first = David

| doi = 10.1016/j.jnt.2008.04.007

Line 48 ⟶ 43:

| 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.

*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.

Line 63 ⟶ 55:

==Notes==

{{reflist}}