NÚMEROS INTEIROS COMO SOMA DE QUADRADOS

◆ú♠❡r♦s ■♥t❡✐r♦s ❈♦♠♦ ❙♦♠❛ ❞❡

◗✉❛❞r❛❞♦s

♣♦r

❏♦ã♦ ❊✈❛♥❣❡❧✐st❛ ❈❛❜r❛❧ ❞♦s ❙❛♥t♦s

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

❆❣r❛❞❡❝✐♠❡♥t♦s

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❢❛③❡r ✉♠❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✜❝❛ s♦❜r❡ ♦ t❡♠❛ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✐♥t❡✐r♦s ❝♦♠♦ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s✱ ♣❛r❛ ♦s ❝❛s♦s ♦♥❞❡ t❡♠♦s s♦♠❛ ❞❡ ❞♦✐s✱ três ❡ q✉❛tr♦ q✉❛❞r❛❞♦s✳ ❆ ✐❞❡✐❛ é ❡st✉❞❛r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❣❛r❛♥t✐r q✉❛♥❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♣♦❞❡rá s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s ❡ q✉❛tr♦ q✉❛❞r❛❞♦s✳ ❖ ❢♦❝♦ ❝❡♥tr❛❧ ❡stá ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦s q✉❛tr♦ q✉❛❞r❛❞♦s ❞❡ ▲❛❣r❛♥❣❡✱ ❛♣❡s❛r ❞❡ t❡r♠♦s ✐❞♦ ✉♠ ♣♦✉❝♦ ❛❞✐❛♥t❡ ❡st✉❞❛♥❞♦ ❛ té❝♥✐❝❛ ❞♦ ❞❡s❝❡♥s♦ ✐♥✜♥✐t♦ ❞❡ ❋❡r♥❛t ❡ ♦ ❝❛s♦ ♥❂✸ ❞♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳ P♦r ✜♠✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ q✉❡ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♥❛s sér✐❡s ✜♥❛✐s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❝✉❥♦ ❝♦♥t❡ú❞♦ ❛❜♦r❞❛❞♦ ♥❡st❛ s❡q✉ê♥❝✐❛ sã♦ ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❞♦ ❝❛♣ít✉❧♦ 2 q✉❡ r❡♠❡t❡ ❛

r❡♣r❡s❡♥t❛çã♦ ❞❡ ✐♥t❡✐r♦s ❝♦♠♦ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s✳

P❛❧❛✈r❛s ❈❤❛✈❡✿ ◆ú♠❡r♦s ✐♥t❡✐r♦s✱ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ s♦♠❛ ❞❡ q✉❛✲ ❞r❛❞♦s✳

❆❜str❛❝t

❚❤✐s ♣❛♣❡r ✐s ❛ s✉r✈❡② ♦♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✐♥t❡❣❡rs ❛s s✉♠s ♦❢ sq✉❛r❡s ❢♦r t❤❡ ❝❛s❡s ✇❤❡r❡ ✇❡ ❤❛✈❡ t❤❡ s✉♠ ♦❢ t✇♦✱ t❤r❡❡ ❛♥❞ ❢♦✉r sq✉❛r❡s✳ ❚❤❡ ✐❞❡❛ ✐s t♦ st✉❞② ❝♦♥❞✐t✐♦♥s s♦ t❤❛t ✇❡ ❝❛♥ ❡♥s✉r❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♥✉♠❜❡rs t❤❛t ❛r❡ ✇r✐tt❡♥ ❛s t❤❡ s✉♠ ♦❢ t✇♦ ❛♥❞ ❢♦✉r sq✉❛r❡✳ ❚❤❡ ❝❡♥tr❛❧ ❢♦❝✉s ✐s t❤❡ st❛t❡♠❡♥t ♦❢ t❤❡ t❤❡♦r❡♠ ♦❢ ▲❛❣r❛♥❣❡ ❢♦✉r sq✉❛r❡s✱ ❛❧t❤♦✉❣❤ ✇❡ ❤❛✈❡ ❣♦♥❡ ❛ ❧✐tt❧❡ ❢✉rt❤❡r st✉❞②✐♥❣ ❋❡r♠❛t✬ s t❡❝❤♥✐q✉❡ ♦❢ ✐♥✜♥✐t❡ ❞❡s❝❡♥s❡ ❛♥❞ t❤❡ ❝❛s❡ ♥ ❂ ✸ ♦❢ ❋❡r♠❛t✬s ❧❛st t❤❡♦r❡♠✳ ❋✐♥❛❧❧②✱ ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❞✐❞❛❝t✐❝ s❡q✉❡♥❝❡ t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ✜♥❛❧ ❣r❛❞❡s ♦❢ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ ❛♥❞ ♠✐❞❞❧❡ s❝❤♦♦❧✱ ❛❞❞r❡ss✐♥❣ ❈❤❛♣t❡r ✷ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥✳

❑❡②✇♦r❞s✿ ❲❤♦❧❡ ♥✉♠❜❡rs✱ ❋❡r♠❛t✬s ❧❛st t❤❡♦r❡♠✱ t❤❡ s✉♠ ♦❢ sq✉❛r❡s✳

❙✉♠ár✐♦

✶ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ■♠♣♦rt❛♥t❡s ✶

✶✳✶ ❘❡sí❞✉♦s ◗✉❛❞rát✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ■♥t❡✐r♦s ❝♦♠♦ ❙♦♠❛ ❞❡ ◗✉❛❞r❛❞♦s ✶✸ ✷✳✶ ❖ Pr♦❜❧❡♠❛ ❞❡ ❲❛r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❙♦♠❛ ❞❡ ❞♦✐s ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❙♦♠❛ ❞❡ ❚rês ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❙♦♠❛ ❞❡ ◗✉❛tr♦ ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺ ❯♠ ❚❡♦r❡♠❛ ❞❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✻ ❉❡s❝❡♥s♦ ■♥✜♥✐t♦ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✼ ❖ Ú❧t✐♠♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✸ ❯♠❛ Pr♦♣♦st❛ ❞❡ ❆t✐✈✐❞❛❞❡ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✹✾

✸✳✶ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❆t✐✈✐❞❛❞❡ Pr♦♣♦st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷ ❙♦❧✉çã♦ ❡ ❈♦♠❡♥tár✐♦ ❞❡ ❝❛❞❛ ■t❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

❆ ❘❡s✉❧t❛❞♦s ❈♦♠♣❧❡♠❡♥t❛r❡s ✺✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾

■♥tr♦❞✉çã♦

❆ ✐❞❡✐❛ ❞❡ r❡♣r❡s❡♥t❛r ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❝♦♠♦ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s s✉r❣❡ ♥❛✲ t✉r❛❧♠❡♥t❡ ❛♦ t❡♥t❛r♠♦s ❡♥❝♦♥tr❛r tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❞❡ ❧❛❞♦s ✐♥t❡✐r♦s✳ ➱ ✉♠ ♣r♦❜❧❡♠❛ ❛♥t✐❣♦ ❡ ✉♠ ❞♦s ♣r✐♠❡✐r♦s ❛ ❡st✉❞á✲❧♦ ❢♦✐ ❉✐♦❢❛♥t♦ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ♦ q✉❛❧ ❡s❝r❡✈❡ ❡♠ s✉❛ ♦❜r❛ ♣r✐♠❛ ✐♥t✐t✉❧❛❞❛ ❛r✐t♠ét✐❝❛ ✳ ❙é❝✉❧♦s ♠❛✐s t❛r❞❡ ♦ ♠❛t❡♠át✐❝♦ ❝❤❛♠❛❞♦ ❇❛❝❤❡t ❢❛③ ❛ tr❛❞✉çã♦ ❞❛ ♦❜r❛ ❞❡ ❉✐♦❢❛♥t♦ ♣❛r❛ ♦ ❧❛t✐♠ ❡ ♣♦r ✐ss♦ ❡st❡ ♣r♦❜❧❡♠❛ ❢♦✐ ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝♦♥❥❡❝t✉r❛ ❞❡ ❇❛❝❤❡t✳ ▼❛s ❢♦✐ ❊❞✉❛r❞ ❲❛r✐♥❣ q✉❡ ❢❡③ ✈ár✐❛s ❛✜r♠❛çõ❡s s♦❜r❡ ❡st❡ t❡♠❛ ✐♥❝❧✉s✐✈❡ q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦ q✉❛tr♦ q✉❛❞r❛❞♦s✳ ▼❛t❡♠át✐❝♦s ❞❡ ✈ár✐❛s é♣♦❝❛s ♠♦str❛r❛♠ ✐♥t❡r❡ss❡ ❡♠ ❞❡♠♦♥str❛r ❡st❡ ❡ ♦✉tr♦s r❡s✉❧t❛❞♦s q✉❡ ❲❛✲ r✐♥❣ ❤❛✈✐❛ ❡♥✉♥❝✐❛❞♦✱ ❡♥tr❡ ❡❧❡s✱ ❋❡r♠❛t ❡ ▲❛❣r❛♥❣❡✱ ❡ ✐st♦ ❣❡r♦✉ ♠✉✐t❛ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ❞❛ é♣♦❝❛✳ ▼❛s✱ ❢♦✐ ❛♣❡♥❛s ♥♦ ❛♥♦ ❞❡ 1909 q✉❡ ♦ ♠❛t❡♠át✐❝♦

❍✐❧❜❡rt ❞❡♠♦♥str♦✉ q✉❡ ♣❛r❛ ❝❛❞❛ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ s✱ ❤á ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ g(s)✱

q✉❡ ✐♥❞❡♣❡♥❞❡ ❞❡ n✱ t❛❧ q✉❡ n ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❝♦♠♦ ❛ s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦ g(s)

s✲és✐♠❛s ♣♦tê♥❝✐❛s ♣♦s✐t✐✈❛s✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛ t❡♦r✐❛ ❞♦s r❡sí❞✉♦s q✉❛✲ ❞rát✐❝♦s✱ ❞❡✜♥✐♥❞♦ ❡ ❞❡♠♦♥str❛♥❞♦ r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ❛♥❞❛♠❡♥t♦ ❞❡st❛ ♣❡sq✉✐s❛✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ tr❛t❛♠♦s ❞♦ t❡♠❛ ❝❡♥tr❛❧ ❞❛ ♣❡sq✉✐s❛ q✉❡ é ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✐♥t❡✐r♦s ❝♦♠♦ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s✳ ◆ã♦ ❢❛r❡♠♦s ❛q✉✐ ✉♠ ❡st✉❞♦ ❛♣r♦❢✉♥❞❛❞♦ s♦❜r❡ ❡st❡ t❡♠❛✱ tr❛t❛r❡♠♦s ❛♣❡♥❛s ❞♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ♣❛r❛ ❛ s♦♠❛ ❞❡ ❞♦✐s✱ três ❡

❛♦ ♣r♦♣ós✐t♦✳ ❱❡r❡♠♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ♥ú♠❡r♦s ✐♥t❡✐r♦s q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s ❡ q✉❛tr♦ q✉❛❞r❛❞♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ❢❛❧❛r❡♠♦s ❞♦s ❞♦✐s r❡s✉❧t❛❞♦s ❝❡♥tr❛✐s ❞❡st❡ tr❛❜❛❧❤♦ q✉❡ sã♦✿ ♦ t❡♦r❡♠❛ ❞♦s q✉❛tr♦ q✉❛❞r❛❞♦s ❞❡ ▲❛❣r❛♥❣❡ ❡ ♦ t❡♦r❡♠❛ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❊✉❧❡r✳ ❋♦♠♦s ✉♠ ♣♦✉❝♦ ♠❛✐s ❛❞✐❛♥t❡ ❡ ❛✐♥❞❛ ✜③❡♠♦s ❞✉❛s s❡çõ❡s ❜❡♠ ✐♥t❡r❡ss❛♥t❡s✿ ✉♠❛ s♦❜r❡ ❛ té❝♥✐❝❛ ❞♦ ❞❡s❝❡♥s♦ ✐♥✜♥✐t♦ ❞❡ ❋❡r♠❛t✱ ♦♥❞❡ ✜③❡♠♦s ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ♣♦❞❡r♠♦s ❝♦♠♣r❡❡♥❞❡r ♠❡❧❤♦r s✉❛ ✉t✐❧✐③❛çã♦✱ ♥❛ ♦✉tr❛ s❡çã♦✱ r❡❧❡♠❜r❛♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ út❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ❡ ✜♥❛❧✐③❛♠♦s ❢❛③❡♥❞♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♠❡s♠♦✱ ♦ ❝❛s♦n = 3✱

♣❛r❛ t❡r♠♦s ♠❛✐s ♦✉ ♠❡♥♦s ❛ ✐❞❡✐❛ ❞❡ ❝♦♠♦ é ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❚❡♦r❡♠❛✳ ◆♦ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❡❧❛❜♦r❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❜❛s❡❛❞❛ ♥❛ t❡♦r✐❛ ❡①♣♦st❛ ♥♦ ❝❛♣ít✉❧♦ 2✳ ❊❧❛ ❡stá ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ♣❛rt❡s✱ ❛ ♣r✐♠❡✐r❛ ❛❜♦r❞❛

♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞♦ ❝❛♣ít✉❧♦ 2✱ ❡♥q✉❛♥t♦ ❛ s❡❣✉♥❞❛ ♣❛rt❡ é ✉♠❛ ❛♣❧✐❝❛çã♦

❛ ❣❡♦♠❡tr✐❛ ❞❡st❡s ❝♦♥❤❡❝✐♠❡♥t♦s✳ ❆ ❛t✐✈✐❞❛❞❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ♥❛s sér✐❡s ✜♥❛✐s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ■■ ❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣♦❞❡♥❞♦ t❡r ót✐♠♦ r❡♥❞✐♠❡♥t♦ ❡♥tr❡ ♦s ❛❧✉♥♦s ✈✐st♦ q✉❡ ❡❧❛ ✈❛✐ ❞❡ ✉♠ ♥í✈❡❧ ♠❛✐s ❡❧❡♠❡♥t❛r ♣❛r❛ ♦ ♥í✈❡❧ ♠❛✐s ❝♦♠♣❧❡①♦✳

❈❛♣ít✉❧♦ ✶

❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ■♠♣♦rt❛♥t❡s

◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ♥♦ ❡st✉❞♦ ❞♦s r❡sí❞✉♦s q✉❛❞rát✐❝♦s✱ ❡♥✉♥❝✐❛♥❞♦ ❡ ❞❡♠♦♥str❛♥❞♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s q✉❡ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ r❡s✉❧t❛❞♦s ♣♦st❡r✐♦r❡s✳

✶✳✶ ❘❡sí❞✉♦s ◗✉❛❞rát✐❝♦s

❖ ✐♥t❡r❡ss❡ ♠❛✐♦r ♥♦ ❡st✉❞♦ ❞♦s r❡sí❞✉♦s q✉❛❞rát✐❝♦s ❡stá ❡♠ ❡st✉❞❛r ❛s s♦❧✉çõ❡s ♣❛r❛ ❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ a (mod m)✳ ◗✉❛♥❞♦ m é ✉♠ ♣r✐♠♦ í♠♣❛r ❡ (a, m) = 1

✭(a, b) é ❛ ♥♦t❛çã♦ ♣❛r❛ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡a ❡ b✮✱ ❛ ❝♦♥❣r✉ê♥❝✐❛✱ ❝❛s♦

t❡♥❤❛ s♦❧✉çã♦✱ t❡rá ❡①❛t❛♠❡♥t❡ ❞✉❛s s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s✱ é ♦ q✉❡ ♠♦str❛r❡♠♦s ♥♦ t❡♦r❡♠❛ ❛❜❛✐①♦✳

❚❡♦r❡♠❛ ✶✳✶ P❛r❛ p♣r✐♠♦ í♠♣❛r ❡a✉♠ ✐♥t❡✐r♦ ♥ã♦ ❞✐✈✐sí✈❡❧ ♣♦rp✱ ❛ ❝♦♥❣r✉ê♥❝✐❛

❛❜❛✐①♦✱ ❝❛s♦ t❡♥❤❛ s♦❧✉çã♦✱ t❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p✳

x2

≡a (mod p)

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ x1 s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡

−x1 t❛♠❜é♠ é s♦❧✉çã♦ ♣♦✐s✱ (−x1)2 = (x1)2 ≡ a (mod p)✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡

❡st❛s s♦❧✉çõ❡s sã♦ ✐♥❝♦♥❣r✉❡♥t❡s✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ x1 ❡ −x1 s❡❥❛♠

❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦p✱ ♦✉ s❡❥❛✱ x1 ≡ −x1 (mod p)✱ ❞❛íx1+x1 ≡ −x1+x1 (mod p)

♣♦rt❛♥t♦✱ 2x1 ≡0 (mod p)✳ ❚❡♠♦s q✉❡pé í♠♣❛r ❡ ♥ã♦ ❞✐✈✐❞❡x1 ❡ s❛❜❡♥❞♦ q✉❡x1 é

❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ♦❝♦rr❡r ❛ ❝♦♥❣r✉ê♥❝✐❛2x1 ≡0

(mod p)✱ ♣♦✐s p♥ã♦ ❞✐✈✐❞❡ ❛ ❡ ❛❧é♠ ❞✐ss♦x2

1 ≡a (mod p)❞❛í ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡

p ♥ã③♦ ❞✐✈✐❞❡ x2

1 ❡ ♣♦rt❛♥t♦ ♥ã♦ ❞✐✈✐❞❡ x1✱ ❛ss✐♠ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ x1 ❡ −x1

sã♦ ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉♦ p✳ ❆ ♥♦ss❛ ♠❡t❛ ❛❣♦r❛ é ♠♦str❛r q✉❡ ❡①✐st❡♠ ❛♣❡♥❛s

❡st❛s ❞✉❛s s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p✳ ❆ss✐♠✱ s❡❥❛ y ✉♠❛ s♦❧✉çã♦ ❞❡ x2

≡ a

(mod p)✱ ❡♥tã♦ y2

≡ a (mod p)✱ ❝♦♠♦ x1 é s♦❧✉çã♦ t❡r❡♠♦s q✉❡ x21 ≡ a (mod p)✱

♣♦rt❛♥t♦x2

1 ≡y

2

≡a (mod p)❡ ❛ss✐♠✱x2 1−y

2

≡0 (mod p)✱ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r (x1+y)(x1−y)≡0 (mod p)✱ ❝♦♠♦ pé ♣r✐♠♦ t❡♠♦s q✉❡ p|x1+y ♦✉ p|x1−y✱ ♦

q✉❡ é ♦ ♠❡s♠♦ q✉❡x1+y≡0 (mod p)♦✉x1−y≡0 (mod p)❞❛íy≡ −x1 (mod p)

♦✉y_≡x1 (mod p)✳ P♦rt❛♥t♦✱ ❝❛s♦ ❡①✐st❛ s♦❧✉çõ❡s✱ só ❡①✐st❡♠ ❛♣❡♥❛s ❞✉❛s s♦❧✉çõ❡s

✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p✳

❉❡✜♥✐çã♦ ✶✳✶ ❖ ❝♦♥❥✉♥t♦ A =_{r1, r2, . . . , rs} é ✉♠ s✐st❡♠❛ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ p

s❡✿

✶✳ ri ♥ã♦ ❢♦r ❝♦♥❣r✉❡♥t❡ ❛ rj ♠ó❞✉❧♦ p ♣❛r❛ i6=j

✷✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ n✱ ❡①✐st❡ ✉♠ ri t❛❧ q✉❡ n≡ri (mod p)✳

❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠a❡p✐♥t❡✐r♦s ❝♦♠(a, p) = 1✳ ❉✐③❡♠♦s q✉❡aé r❡sí❞✉♦ q✉❛❞rá✲

t✐❝♦ ♠ó❞✉❧♦ p s❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ a (mod p) t✐✈❡r s♦❧✉çã♦✳ ❈❛s♦ ❛ ❝♦♥❣r✉ê♥❝✐❛

♥ã♦ t❡♥❤❛ s♦❧✉çã♦✱ ❞✐③❡♠♦s q✉❡ a ♥ã♦ é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦p ♦✉ q✉❡ a é ✉♠

r❡sí❞✉♦ ♥ã♦✲q✉❛❞rát✐❝♦✳

❚❡♦r❡♠❛ ✶✳✷ ❙❡❥❛ p ✉♠ ♣r✐♠♦ í♠♣❛r✳ ❉❡♥tr❡ ♦s ♥ú♠❡r♦s _{1, 2, 3, . . . , p₋1_}✱

✈❡❥❛ q✉❡ p−1

2 sã♦ r❡sí❞✉♦s q✉❛❞rát✐❝♦s ❡ p−1

2 ♥ã♦ sã♦✳

✶✳✶✳ ❘❊❙❮❉❯❖❙ ◗❯❆❉❘➪❚■❈❖❙

❉❡♠♦♥str❛çã♦✿

❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦s q✉❛❞r❛❞♦s ❞♦s ♥ú♠❡r♦s ❞❡ 1 ❛ p₋1✳ ❆ss✐♠✱ (1)2

≡ 1 (mod p)✱ ♦✉ s❡❥❛✱ 1 é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ 1 (mod p)✱ ♠❛s ♦❜✲

s❡r✈❡♠♦s q✉❡ (₋1)2

= (1)2

≡ 1 (mod p)✱ ♦✉ s❡❥❛✱ ₋1 t❛♠❜é♠ é s♦❧✉çã♦ ❞❡st❛

❝♦♥❣r✉ê♥❝✐❛ ❡✱ ❛❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ −1 _≡ p+ (₋1) = p₋1 (mod p)✱ ♦♥❞❡ p₋1

t❛♠❜é♠ é s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛✱ ♣♦✐s (p₋1)2

=p2

−2p+ 1✱ ♣♦rt❛♥t♦(p₋1)2

≡1 (mod p)✱ ❧♦❣♦ ♣❡❧♦ t❡♦r❡♠❛ 1.1 ❝♦♥❝❧✉í♠♦s q✉❡ 1 ❡ p₋1 sã♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s

✐♥❝♦♥❣r✉❡♥t❡s ❞❡ x2

≡1 (mod p)✱ ❡♥tr❡ ♦s ♥ú♠❡r♦s 1,2,3, . . . , p₋1✳

❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ♦ 22 _{q✉❡ s❡rá ❝♦♥❣r✉❡♥t❡ ❛ ❛❧❣✉♠ ♥ú♠❡r♦}

k ❞✐❢❡r❡♥t❡ ❞❡

1✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ (₋2)2 t❛♠❜é♠ ♦ é✳ ❖❜s❡r✈❛♥❞♦ q✉❡

−2 _≡ p+ (₋2) = p₋2 (mod p)✱ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ t❡♦r❡♠❛1.1❝♦♥❝❧✉í♠♦s q✉❡2❡p₋2sã♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s

✐♥❝♦♥❣r✉❡♥t❡s ❞❡ x2

≡k (mod p) ❞❡♥tr❡ ♦s ♥ú♠❡r♦s i= 1,2,3, . . . , p₋1✳

❙❡ t♦♠❛r♠♦s ❛❣♦r❛ 32 _{❡ ❡st❡ s❡rá ❝♦♥❣r✉❡♥t❡ ❛ ❛❧❣✉♠}

q ❞✐❢❡r❡♥t❡ ❞❡ 1 ❡ ❞❡ k✱

❛♥❛❧❛❣♦♠❡♥t❡ ❛♦ q✉❡ ❢♦✐ ♠♦str❛❞♦ t❡♠♦s q✉❡ (₋3)2 t❛♠❜é♠ s❡rá ❝♦♥❣r✉❡♥t❡ ❛

q ❡

❛❧é♠ ❞✐ss♦✱−3_≡p₋3 (mod ✮ ❡♥tã♦₋3❡p₋3sã♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s

❞❡ x2

≡q (mod p) ❞❡♥tr❡ ♦s ♥ú♠❡r♦s i= 1,2,3, . . . , p₋1✳

❚❡♠♦s ❝♦♠♦ r❡sí❞✉♦s q✉❛❞rát✐❝♦s ♦s ♥ú♠❡r♦s1✱ k ❡ q ❞❛s ❝♦♥❣r✉ê♥❝✐❛s x2

≡ 1 (mod p)✱ x2

≡k (mod p)❡x2

≡q (mod p)s❡♥❞♦ s✉❛s r❡s♣❡❝t✐✈❛s s♦❧✉çõ❡s ♦s ♣❛r❡s (1, p₋1)✱(2, p₋2)❡(3, p₋3)✳ ❙❡ ❝♦♥t✐♥✉❛r♠♦s ♣r♦❝❡❞❡♥❞♦ ❞❡st❛ ♠❛♥❡✐r❛ t❡r❡♠♦s

p−1

2 ♣❛r❡s ❞❡ s♦❧✉çõ❡s

(1, p₋1),(2, p₋2),(3, p₋3), . . . ,

p₋1 2 ,

p₋1 2

♦♥❞❡ ❝❛❞❛ ♣❛r é s♦❧✉çã♦ ♣❛r❛ ✉♠❛ ❞❡♥tr❡ ❛s p−1

2 ❝♦♥❣r✉ê♥❝✐❛s ❛ss♦❝✐❛❞❛s ❛ p−1

2

r❡sí❞✉♦s q✉❛❞rát✐❝♦s✳

❚❡♦r❡♠❛ ✶✳✸ P❛r❛ p ♣r✐♠♦✱ ❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ −1 (mod p) t❡♠ s♦❧✉çã♦ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ p= 2 ♦✉ p_≡1 (mod 4)✳

❉❡♠♦♥str❛çã♦✿

❈❛s♦ ♣❂✷✿ ❞❡ ❢❛t♦✱ ♣❛r❛ x = 1 ❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ −1 (mod 2) t❡♠ s♦❧✉çã♦✱

s❛❜❡♠♦s q✉❡2_≡0 (mod 2)✱ ❞❛í ❛❞✐❝✐♦♥❛♥❞♦₋1❛ ❝♦♥❣r✉ê♥❝✐❛✱ ♦❜t❡♠♦s2 + (₋1)_≡ 0 + (₋1) (mod 2)❛ss✐♠✱ 1_{≡ −}1 (mod 2) ❡ ❞❛í12

≡ −1 (mod 2)✱ ♦ q✉❡ ♥♦s ♠♦str❛

q✉❡ r❡❛❧♠❡♥t❡ x= 1 é s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛✳ ❘❡st❛ ❛❣♦r❛ ♠♦str❛r q✉❡ ❡①✐st❡ ✉♠❛

s♦❧✉çã♦ ♣❛r❛ p_≡1 (mod 4)✳

❙❡♥❞♦ p ♣r✐♠♦ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✱ ✈✐❞❡ ❛♣ê♥❞✐❝❡✱ ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡

(p₋1)! _{≡ −}1 (mod p)✱ ❝♦♠♦ p > 2 é ♣r✐♠♦ ❡♥tã♦ p₋1 é ♣❛r✱ ❧♦❣♦ (p₋1)! t❡♠

✉♠❛ q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ❢❛t♦r❡s✱ ♦✉ s❡❥❛✱ p₋1 ❢❛t♦r❡s ❡①❛t❛♠❡♥t❡✳ ❉❛í ♣♦❞❡r❡♠♦s

❡s❝r❡✈❡r ♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

(p₋1)! = (p₋1)_·(p₋2)_·. . .(p₋k). . .

p+ 1 2

!_{≡ −}1 (mod p)✱

♦❜s❡r✈❡♠♦s q✉❡ ❤á ♥❡st❡ ♠♦♠❡♥t♦ p−1

2 ❢❛t♦r❡s✱ ❞❡ ❢❛t♦✱ ♦❜s❡r✈❡♠♦s q✉❡ ♦s ❢❛t♦r❡s

((p₋1),(p₋2), . . .),(p₋k), . . .3,2,1) ❢♦r♠❛♠ ✉♠❛ P✳❆ ❞❡ r❛③ã♦₋1✱ ❞❛í ♦ t❡r♠♦

ap−1

2 = (p−1) +

p₋1

2 −1

(₋1) =p₋1 + 1₋ 1−p

2 =p₋ 1−p

2 = 2p+ 1−p

2 = p+ 1

2 .

❆✐♥❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

(p₋1)! = (p₋1)_·(p₋2)_·. . ._·(p₋k)_·. . ._·

p+ 1 2

!_{≡ −}1 (mod p)

❝♦♠♦✱

✶✳✶✳ ❘❊❙❮❉❯❖❙ ◗❯❆❉❘➪❚■❈❖❙

((p₋1)_·(p₋2)_·. . ._·(p₋k)_·. . ._·

p+ 1 2

)_· (

p₋1

2

. . . k . . .4_·3_·2_·1)_{≡ −}1 (mod p). ✭✶✳✶✮

❖❜s❡r✈❡♠♦s q✉❡(p₋1)! ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s✱ ♦♥❞❡ ❝❛❞❛ ✉♠❛ t❡♠ p−1 2

❢❛t♦r❡s✳ P♦❞❡r❡♠♦s r❡❡s❝r❡✈❡r ❛❣r✉♣❛♥❞♦ ♦s ❢❛t♦r❡s ❛♦s ♣❛r❡s✱ ❞❛í ✜❝❛r❡♠♦s ❝♦♠✱

1_·(p₋1)_·2_·(p₋2)_·. . ._·k(p₋k)_·. . ._·(p−1 2 )·(

p+1

2 )≡ −1 (mod p)✳ ◆♦t❡ q✉❡ ❛✐♥❞❛

♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ❝♦♠♦ ♦ ♣r♦❞✉tór✐♦✱ ❛❜❛✐①♦✿

p−1 2

Y

k=1

k(p₋k)_{≡ −}1 (mod p). ✭✶✳✷✮

❋❛ç❛♠♦s ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✱ k(p₋k)_{≡ −}k2

(mod p)✱ q✉❡ é ❞❡ ❢á❝✐❧ ❥✉st✐✜❝❛✲

t✐✈❛✱ ♣♦✐s

n=k(p₋k) =kp₋k2

=kp+ (₋k2

) =k(p₋k)_{≡ −}k2

(mod p),

❛ss✐♠✱

Qp−1₂

k=1k(p−k)≡

Qp−1₂

k=1(−k 2

)_{≡ −}1 (mod p)✱

♣♦rt❛♥t♦ Qp−1₂

k=1(−k 2

)_{≡ −}1 (mod p)✱ ♥♦t❡ q✉❡

p−1 2

Y

k=1

(₋k2

) = (₋12

)_·(₋22

). . ._·(₋

p₋1

2

2

) = (₋1)_·(₋1)_·. . ._·(₋1)(12

)_·(22

). . ._·

p₋1 2

2

= (₋1)p−12

1_·2. . ._· p−1

2

2

= (₋1)p−12

  p−1 2 Y k=1 k   2

≡ −1 (mod p)). ✭✶✳✸✮

❈♦♠♦ p _≡ 1 (mod 4)✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ p−21 é ♣❛r✳ ❉❡ ❢❛t♦✱ s❡♥❞♦ p ≡ 1

(mod 4) ❡①✐st❡ s✐♥t❡✐r♦ t❛❧ q✉❡ p= 4s+ 1❧♦❣♦ p₋1 = 4s✱ s❡♥❞♦ p✉♠ ♣r✐♠♦ ♠❛✐♦r

❞♦ q✉❡ ❞♦✐s ❡♥tã♦ ❡st❡ é ✐♠♣❛r✱ ♣♦rt❛♥t♦ p₋1 é✱ ♣❛r✱ ❡♥tã♦ ❛♦ ❞✐✈✐❞✐r♠♦s ❛♠❜♦s

♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ♣♦r 2 t❡r❡♠♦s p−1

2 = 2s✱ ♦ q✉❡ ♥♦ ❞✐③ q✉❡ p−1

2 é ♣❛r✳ ❉❛í✱

(₋1)p−12 = 1✱ ❧♦❣♦✱ (Q p−1

2

k=1k) 2

≡ −1 (mod p) ♦ q✉❡ ♥♦s ❞✐③ q✉❡

x=Qp−1₁

k=1 = 1·2·3·. . .·

p₋1 2 =

p₋1

2

!

é ✉♠❛ s♦❧✉çã♦ ❞❡ x2

≡ −1 (mod p)✳ ❱❛♠♦s s✉♣♦r ❛❣♦r❛ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛

x2

≡ −1 (mod p) t❡♥❤❛ s♦❧✉çã♦ ❡ q✉❡ p > 2✱ ♣♦✐s x2

≡ −1 (mod 2) t❡♠ s♦❧✉çã♦

x= 1✳ ❊❧❡✈❛♥❞♦ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❛ ♣♦tê♥❝✐❛ p−21 ♦❜t❡♠♦s

(x2

)p−12 ≡(−1) p−1

2 (mod p) q✉❡ é ♦ ♠❡s♠♦ q✉❡

xp−1

≡(₋1)p−12 (mod p) ✳

❈♦♠♦ x2

≡ −1 (mod p)✱ ♥ós ♣♦❞❡♠♦s ❞✐③❡r q✉❡ p ∤ x2 ❡ ❞❛í

p ∤ x✱ ♣♦rt❛♥t♦

♣❡❧♦ ♣❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ✈✐❞❡ ❛♣❡♥❞✐❝❡✱ (x)p−1

≡ 1 (mod p)✱ ❛í t❡r❡♠♦s

✶✳✶✳ ❘❊❙❮❉❯❖❙ ◗❯❆❉❘➪❚■❈❖❙

(₋1)p−12 ≡1 (mod p) ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❛✜r♠❛r q✉❡ p−1

2 é ♣❛r✱ ❞❛í ❡①✐st❡ j ✐♥t❡✐r♦

t❛❧ q✉❡ p−1

2 = 2j✱ ♦ q✉❡ ♣♦❞❡♠♦s ❛✐♥❞❛ ❝♦♠♦ p−1 = 4j ❡ ❛ss✐♠ t❡r♠♦sp= 4j+ 1♦

q✉❡ ❛❝❛rr❡t❛ p_≡1 (mod 4)✱ ❡ ❛ss✐♠ ❝♦♥❝❧✉í♠♦s ❛ ♥♦ss❛ ❞❡♠♦♥str❛çã♦✳

❉❡✜♥✐çã♦ ✶✳✸ P❛r❛p✉♠ ♣r✐♠♦ í♠♣❛r ❡ a✉♠ ✐♥t❡✐r♦ ♥ã♦ ❞✐✈✐sí✈❡❧ ♣♦rp✱ ❞❡✜♥✐♠♦s

♦ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡ (a p) ♣♦r✿

a p

=







1, s❡ ❛ é ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦ ❞❡ ♣❀

−1, s❡ ❛ ♥ã♦ é ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦ ❞❡ ♣✳

❚❡♦r❡♠❛ ✶✳✹ ✭❈r✐tér✐♦ ❞❡ ❊✉❧❡r✮ ❙❡ p ❢♦r ✉♠ ♣r✐♠♦ í♠♣❛r ❡ a ✉♠ ✐♥t❡✐r♦ ♥ã♦✲

❞✐✈✐sí✈❡❧ ♣♦r p✱ ❡♥tã♦✿

a p

≡ap−12 (mod p) ❉❡♠♦♥str❛çã♦✿

❙✉♣♦♥❞♦ q✉❡✱(a

p) = 1✱ ♦✉ s❡❥❛✱ ❛ ❝♦♥❣r✉ê♥❝✐❛x 2

≡a (mod p)t❡♠ s♦❧✉çã♦✳ ❙❡❥❛

y t❛❧ s♦❧✉çã♦✱ ❞❛í t❡r❡♠♦s q✉❡ y2

≡a (mod p) ✐♠♣❧✐❝❛♥❞♦ ❡♠ y2

−a _≡0 (mod p)✱

❛ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡p❞✐✈✐❞❡y2

−a✱ ♠❛sp♥ã♦ ❞✐✈✐❞❡a✱ ♣♦rt❛♥t♦ ♥ã♦ ♣♦❞❡ ❞✐✈✐❞✐r y✱ ❧♦❣♦ (y, p) = 1 ❡ ♣❡❧♦ ♣❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t t❡♠♦s q✉❡ yp−1

≡1 (mod p)✱

❛ss✐♠ (y2

)p−12 ≡ a p−1

2 (mod p) ❡♥tã♦ a p−1

2 ≡ yp−1 ≡ 1 (mod p)✱ ♣♦rt❛♥t♦ a p−1

2 ≡ 1

(mod p)❡ ❛ss✐♠ (a p)≡a

p−1

2 ≡1 ❡ ✐st♦ ❝♦♥❝❧✉✐ ♦ ❝❛s♦ ❡♠ q✉❡ (a

p) = 1✳

❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛❣♦r❛ ♦ ❝❛s♦ ❡♠ q✉❡ (a

p) =−1✱ ✐st♦ é✱ t♦♠❡♠♦s a ✉♠ r❡sí❞✉♦

♥ã♦✲q✉❛❞rát✐❝♦ ❞❡p❡ s❡❥❛c✉♠ ❞♦s ✐♥t❡✐r♦s_{1,2,3, . . . , p₋1_}✳ ▲❡♠❜r❛♥❞♦ ✉♠ ♣♦✉❝♦

❞❛s ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦c′

❞❡cx_≡a (mod p)✱ ♦♥❞❡

c′

❡stá ♥♦ ❝♦♥❥✉♥t♦ ♠❡♥❝✐♦♥❛❞♦✳ ❖❜s❡r✈❡♠♦s q✉❡ c′

6

= c✱ ♣♦✐s s❡ c = c′

t❡rí❛♠♦s

c2

≡a (mod p)✱ ♠❛s ✐st♦ ♥♦s ❞✐③ q✉❡ a é r❡sí❞✉♦ q✉❛❞rát✐❝♦✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦

❞❡ q✉❡ (a

p) = −1✳ ❉❛í ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦s ✐♥t❡✐r♦s ❞❡ 1 ❛té p−1 ❡♠ p−1

2 ♣❛r❡s✱ c❡

c′

✱ ♦♥❞❡cc′

≡a (mod p)✱ ♦ q✉❡ ♥♦s ❞á p−1

2 ❝♦♥❣r✉ê♥❝✐❛s✳

c1c

′

1 ≡a (mod p)

c2c

′

2 ≡a (mod p)

✳✳✳ ✳✳✳

cp−1 2 c

′ p−1

2 ≡a (mod p) ▼✉❧t✐♣❧✐❝❛♥❞♦ ♦❜t❡♠♦s

c1c

′

1c2c

′

2. . . cp−1

2 c ′ p−1

2 ≡

ap−12 (mod p) ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛✐♥❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

(p₋1)!_≡ap−12 (mod p) P❡❧♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥ ♦❜t❡♠♦s

ap−12 ≡ −1 (mod p)✱ ❝♦♠♦ q✉❡rí❛♠♦s✳

❚❡♦r❡♠❛ ✶✳✺ ❖ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡ é ✉♠❛ ❢✉♥çã♦ ♠✉❧t✐♣❧✐❝❛t✐✈❛ ❞❡ a✱ ♦✉ s❡❥❛ ✿

ab

p

=

a p

b p

♣❛r❛ a ❡ b ✐♥t❡✐r♦s ♥ã♦✲❞✐✈✐sí✈❡✐s ♣♦r p✳

❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ♦ ❝r✐tér✐♦ ❞❡ ❊✉❧❡r✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✿

✶✳✶✳ ❘❊❙❮❉❯❖❙ ◗❯❆❉❘➪❚■❈❖❙

ab

p

≡(ab)p−12 (mod p) ▲❡♠❜r❛♥❞♦ q✉❡

(ab)p−12 =a p−1 2 b p−1 2 ❡ a p

≡ap−12 (mod p) ❡

b p

≡bp−12 (mod p)✱ ❡ ❛ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡

(ab)p−12 =a p−1 2 b p−1 2 ≡ a p b p

(mod p)✳

P♦rt❛♥t♦✱ ab p = a p b p ✳ ❈♦r♦❧ár✐♦ ✶✳✶ a2 p = 1 ❉❡♠♦♥str❛çã♦✿

❯s❛♥❞♦ ♦ t❡♦r❡♠❛ 1.5 ❡ ❝♦♥s✐❞❡r❛♥❞♦ a = b ❛❧✐❛❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ (a

p) = ±1✱

t❡♠♦s a2 p = a p a p

❝♦♠♦(a_p) =_±1✱ t❡♠♦s q✉❡ s❡ (a_p) = 1✱ ❡♥tã♦

a2 p = a p a p

= 1_·1 = 1

❛❣♦r❛✱ s❡ (a

p) =−1✱ t❡r❡♠♦s

a2

p

=

a p

a p

= (₋1)_·(₋1) = 1

❝♦♥❝❧✉✐♥❞♦ ❛ss✐♠ ❛ ❞❡♠♦♥str❛çã♦✳

❚❡♦r❡♠❛ ✶✳✻ P❛r❛ p ♣r✐♠♦ í♠♣❛r✱ t❡♠♦s✿

−1

p

❂







1, s❡ p_≡1 (mod 4)❀

−1, s❡ p_≡3 (mod 4)✳

❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s ❞♦ ❈r✐tér✐♦ ❞❡ ❊✉❧❡r q✉❡ ✿

−1

p

≡(₋1)p−12 (mod p) ❉❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡(−1

p ) = 1 s❡ p−1

2 ❢♦r ♣❛r ❡ (

−1

p ) =−1

q✉❛♥❞♦ p−1

2 í♠♣❛r✳ ❙❡ p ❢♦r ✉♠ ♣r✐♠♦ í♠♣❛r✱ ❡①✐st❡♠ ❛♣❡♥❛s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s

♣❛r❛ p✱ ❡♠ t❡r♠♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ 4✱ p_≡ 1 (mod 4) ♦✉p _≡3 (mod 4)✳ ❙❡

p _≡ 1 (mod 4)✱ ❡①✐st❡ s ✐♥t❡✐r♦ t❛❧ q✉❡ p = 4s+ 1 ♦♥❞❡ p₋1 = 4s ❡ ❛ss✐♠ t❡r♠♦s

p−1

2 = 2s✱ ♦✉ s❡❥❛✱ p−1

2 é ♣❛r✳ ❙❡ p≡ 3 (mod 4)✱ ❡①✐st❡ k ✐♥t❡✐r♦ t❛❧ q✉❡ p= 4k+ 3

♣♦❞❡♥❞♦ s❡r ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛p₋1 = 2(2k+1)❝♦♥❝❧✉í♥❞♦ q✉❡ p−1

2 = 2k+1✱

♦✉ s❡❥❛✱ p−1

2 é í♠♣❛r✳ P♦rt❛♥t♦✱ q✉❛♥❞♦ p ≡ 1 (mod 4) t❡♠♦s (

−1

p ) = 1 ❡ q✉❛♥❞♦

p_≡3 (mod 4) t❡♠✲s❡(−1

p ) =−1✳

Pr♦♣♦s✐çã♦ ✶✳✶ s❡❥❛♠ a✱ b ❡ m ✐♥t❡✐r♦s t❛✐s q✉❡ m >0 ❡ (a, m) = d✳ ◆♦ ❝❛s♦ q✉❡ d ∤ b ❛ ❝♦♥❣r✉ê♥❝✐❛ ax _≡ b (mod m) ♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠❛ s♦❧✉çã♦ ❡ q✉❛♥❞♦ d _| b

♣♦ss✉✐ ❡①❛t❛♠❡♥t❡ ❞ s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳

❉❡♠♦♥str❛çã♦✿ ❝♦♠♦ a ❡ b sã♦ ✐♥t❡✐r♦s✱ ax _≡ b (mod m) s❡✱ ❡ s♦♠❡♥t❡ s❡✱

❡①✐st✐r y t❛❧ q✉❡ ax =b+ym✱ ♦✉ s❡❥❛✱ b = ax₋ym✳ ❙❛❜❡♠♦s q✉❡ s❡ d∤ b ❡♥tã♦ ❛

✶✳✶✳ ❘❊❙❮❉❯❖❙ ◗❯❆❉❘➪❚■❈❖❙

❡q✉❛çã♦ax₋my =b♥ã♦ t❡♠ s♦❧✉çã♦✱ ❥á s❡d_|bt❡r❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ax₋my =b

♣♦ss✉✐ ✐♥✜♥✐t❛s s♦❧✉çõ❡s q✉❡ sã♦ ❞❛ ❢♦r♠❛x=x0−(m_d)k ❡y=y0−(a_d)k♦♥❞❡(x0, y0)

é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦ ax₋my =b✳ P♦rt❛♥t♦✱ ❛ ❝♦♥❣r✉ê♥❝✐❛ ax _≡b

(mod m) ✐rá ♣♦ss✉✐r ✐♥✜♥✐t❛s s♦❧✉çõ❡s ❞❛❞❛s ♣♦r x =x0−(m_d)✳ ❉❡s❡❥❛♠♦s s❛❜❡r ❛

q✉❛♥t✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s✳ ❉❛í ❡st✉❞❛r❡♠♦s ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ❛s q✉❛✐s

x1 = x0 −(m_d) ❡ x2 = x0 −(m_d) sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳ ❙❡ x1 ❡ x2 ❢♦r❡♠

❝♦♥❣r✉❡♥t❡s ❡♥tã♦ x0−(m_d)k1 ≡x0−(m_d)k2 (mod m)✱ ❛ss✐♠

x0−x0−

m

d

k1 ≡x0−x0−

m

d

k2 (mod m)

❞❛í

−m_dk1 ≡ −

m

d

k2

m

d

k1 ≡

m

d

k2✳

❈♦♠♦ (m

d) |m✱ ❞❡ ❢❛t♦ m =d·( m

d)✱ t❡♠♦s q✉❡ ( m

d, m) = m

d✱ ♣♦rt❛♥t♦ ♣♦❞❡♠♦s

❝❛♥❝❡❧❛r (m

d) ♥❛ ❝♦♥❣r✉ê♥❝✐❛ ❛♥t❡r✐♦r✱ ♣♦rt❛♥t♦ k1 ≡k2 (mod m)✳

❉❛í ❛s s♦❧✉çõ❡s ✐♥❝♦♥❣r✉❡♥t❡s sã♦ ❞❛ ❢♦r♠❛x=x0−(m_d)k✱ ♦♥❞❡ k ♣❡r❝♦rr❡ ✉♠

s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ d✳

❚❡♦r❡♠❛ ✶✳✼ P❛r❛ t♦❞♦ ♣r✐♠♦ p❡①✐st❡♠ ✐♥t❡✐r♦sa✱ b ❡ c✱ ♥ã♦ t♦❞♦s ♥✉❧♦s✱ t❛✐s q✉❡

❛ ❝♦♥❣r✉ê♥❝✐❛ s❡❣✉✐♥t❡ s❡ ✈❡r✐✜❝❛

a2

+b2

+c2

≡0 (mod p)✳

❉❡♠♦♥str❛çã♦✿ P❛r❛p= 2✱ t♦♠❛♥❞♦ a=b= 1❡c= 0✱ t❡r❡♠♦s12

+ 12

+ 02

= 2_≡0 (mod 2)✳ ❆♦ ❝♦♥s✐❞❡r❛r♠♦s p_≡1 (mod 4)t♦♠❛r❡♠♦s b = 1✱ c= 0 ❡ a ❝♦♠♦

s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ −1 (mod p)✳ ❉❛í✱b2

= 12

= 1✱ c2

= 02

= 0

❡ a2

≡ −1 (mod p)✱ ❛ss✐♠✱a2

+b2

+c2

≡ −1 + 1 + 0 = 0 (mod p)✳ ❆❣♦r❛✱ s✉♣♦♥❞♦

q✉❡ p _≡ 3 (mod 4) t♦♠❛r❡♠♦s c = 1 ❡ ✐r❡♠♦s ♠♦str❛r q✉❡ ❡①✐t❡ s♦❧✉çã♦ ♣❛r❛ ❛

❝♦♥❣r✉ê♥❝✐❛

a2

+b2

≡ −1 (mod p)

P❡❧♦ t❡♦r❡♠❛ 1.2✱ s❛❜❡♠♦s q✉❡ ♣❛r❛ ✉♠ ♥ú♠❡r♦ p ♣r✐♠♦ í♠♣❛r t❡r❡♠♦s p−1 2

r❡sí❞✉♦s q✉❛❞rát✐❝♦s ❡ p−1

2 r❡sí❞✉♦s ♥ã♦ q✉❛❞rát✐❝♦s ❞❡♥tr❡ ♦s ♥ú♠❡r♦s 1✱ 2✱ 3✱ . . .✱

p₋1✳ ❊ ❛✐♥❞❛ s❡ q ❢♦r ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦✱ ❡♥tã♦ ❛ ❝♦♥❣r✉ê♥❝✐❛✿ x2

≡q (mod p)

t❡♠ s♦❧✉çã♦ s❡ p ❢♦r ♣r✐♠♦✳ ■r❡♠♦s s✉♣♦r q✉❡d é ♦ ♠❡♥♦r r❡sí❞✉♦ ♣♦s✐t✐✈♦ ♥ã♦✲

q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ p✳ ❙❛❜❡♠♦s q✉❡ 1 é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♣♦✐s✱ 2 _≡ 0 (mod 2) ♦

q✉❡ r❡s✉❧t❛ ❡♠ 1_{≡ −}1 (mod 2)❡ ❛ss✐♠ t❡♠♦s 12

≡ −1 (mod 2)✱ ❡♥tã♦ d_≥2✳ P❡❧♦

t❡♦r❡♠❛ 1.6❝♦♥❝❧✉í♠♦s q✉❡ s❡ p_≡3 (mod 4)❡①✐st❡ k1 ✐♥t❡✐r♦ t❛❧ q✉❡ p= 4k1+ 3❛

q✉❛❧ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝♦♠♦ s❡❣✉❡ p= 4k1+ 3−4 + 4 = 4(k1+ 1)−1❡ ❞❛í p≡ −1

(mod 4)✱ ❡♥tã♦(−1

p ) = −1✱ s❛❜❡♥❞♦ q✉❡d ♥ã♦ é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ❡♥tã♦( d

p) =−1✳

P❡❧♦ t❡♦r❡♠❛ 1.5✱

−d p

=

−1

p d p

= (₋1)(₋1) = 1

❆ ❡①♣r❡ssã♦ ❛❝✐♠❛ ♥♦s ✐♥❢♦r♠❛ q✉❡ −d é ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ p✱ ♦✉

s❡❥❛✱ ❛ ❝♦♥❣r✉ê♥❝✐❛ x2

≡ −d (mod p) t❡♠ s♦❧✉çã♦✳ ❊♥tã♦ s❡❥❛ b t❛❧ q✉❡ b2

≡ −d

(mod p)✳ ❉❡✈❡♠♦s ❡♥❝♦♥tr❛ra❝♦♥✈❡♥✐❡♥t❡ t❛❧ q✉❡a2

≡d₋1 (mod p)✱ ❞❛í✱a2

+b2

≡ −d+d₋1 =₋1 (mod p)✳ ❖❜s❡r✈❡♠♦s q✉❡ a2

≡ d₋1 (mod p) t❡♠ s♦❧✉çã♦✱ ♣♦✐s

d_≥2 ❡d₋1< d s❡♥❞♦d ♦ ♠❡♥♦r r❡sí❞✉♦ ♥ã♦ q✉❛❞rát✐❝♦ ♣♦s✐t✐✈♦ ♠ó❞✉❧♦ pt❡♠♦s

q✉❡a2

≡d₋1 (mod p)t❡♠ s♦❧✉çã♦ ♣♦✐s pé ♣r✐♠♦ ❡d₋1é ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦✳

▲♦❣♦✱

a2

+b2

≡ −1 (mod p)

t❡♠ s♦❧✉çã♦ ❡ ❛ss✐♠✱ ❛ ❝♦♥❣r✉ê♥❝✐❛

a2

+b2

+c2

≡0 (mod p)

é ✈❡r✐✜❝❛❞❛✳

❈❛♣ít✉❧♦ ✷

❘❡♣r❡s❡♥t❛çã♦ ❞❡ ■♥t❡✐r♦s ❝♦♠♦

❙♦♠❛ ❞❡ ◗✉❛❞r❛❞♦s

✷✳✶ ❖ Pr♦❜❧❡♠❛ ❞❡ ❲❛r✐♥❣

❯♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ✧P❛✐ ❞❛ ➪❧✲ ❣❡❜r❛✧❥á ❞❡s❝♦♥✜❛✈❛ q✉❡ t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♣♦❞❡r✐❛♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦ q✉❛tr♦ q✉❛❞r❛❞♦s✳ ❊st❡ ♠❛t❡♠át✐❝♦ ❡r❛ ❉✐♦❢❛♥t♦ ❞❡ ❆❧❡①❛♥❞r✐❛ q✉❡ ♥❛s❝❡✉ ❡♠ 22 ❞❡ ❙❡t❡♠❜r♦ ❞❡ 250 ❛✳❈ ❡ ♠♦rr❡✉ 84 ❛♥♦s ❞❡♣♦✐s✳

❖ ♣r♦❜❧❡♠❛ ✜❝♦✉ ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝♦♥❥❡❝t✉r❛ ❞❡ ❇❛❝❤❡t ♦ q✉❛❧ ❢❡③ ❛ tr❛❞✉çã♦ ♣❛r❛ ♦ ❧❛t✐♠ ❞♦ tr❛❜❛❧❤♦ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❞❡ ❉✐♦❢❛♥t♦ ✐♥t✐t✉❧❛❞♦ ❆r✐t♠é✲ t✐❝❛✳ ▼✉✐t♦s ♠❛t❡♠át✐❝♦s s❡ ✐♥t❡rr❡ss❛r❛♠ ♣♦r ❡st❡ ♣r♦❜❧❡♠❛ ✐♥❝❧✉s✐✈❡ ❋❡r♠❛t✱ ♠❛s t♦❞♦s ♥ã♦ t✐✈❡r❛♠ ê①✐t♦ ❡♠ ❞❡♠♦♥strá✲❧♦✳ ❊♠ ✶✼✼✵ ♦ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês ❊❞✇❛r❞ ❲❛r✐♥❣ ❛✜r♠♦✉ q✉❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦4

q✉❛❞r❛❞♦s✱ ♥♦ ♠á①✐♠♦ 9 ❝✉❜♦s ❡ ♥♦ ♠á①✐♠♦19 q✉❛rt❛s ♣♦tê♥❝✐❛s✳ ❆ ♣❡s❛r ❞❡ ♥ã♦

t❡r ❞❡♠♦♥str❛❞♦ ♥❡♥❤✉♠❛ ❞❡ss❛s ❛✜r♠❛çõ❡s ❡❧❡✱ ❛tr❛✈és ❞❡ ♠✉✐t♦s ❡①❡♠♣❧♦s✱ ❝♦♥✲ ❥❡❝t✉r♦✉ q✉❡ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦s ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦g(s)✱ t❛❧

q✉❡ t♦❞♦ ✐♥t❡✐r♦n ♣♦s✐t✐✈♦ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ ♥♦ ♠á①✐♠♦g(s)s✲és✐♠❛s ♣♦tê♥❝✐❛s

♣♦s✐t✐✈❛s✳

❖ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ❏♦s❡♣❤ ▲♦✉✐s ▲❛❣r❛♥❣❡✱ ❡♠ ✶✼✼✵ ❞❡♠♦♥str❛ q✉❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦ q✉❛tr♦ q✉❛❞r❛❞♦s✱ ❡♠ ✶✽✺✾ é q✉❡ ❢♦✐ ❞❡♠♦♥str❛❞♦ q✉❡ ♦ ❢❛t♦ ❞❡ q✉❡ t♦❞♦ ✐♥t❡✐r♦ é s♦♠❛ ❞❡ ♥♦ ♠á①✐♠♦ ✾ ❝✉❜♦s✳ ◆♦ ❛♥♦ ❞❡ ✶✾✵✾ ♦ ♠❛t❡♠át✐❝♦ ❍✐❧❜❡rt ❞❡♠♦♥str❛ q✉❡ ♣❛r❛ ❝❛❞❛s ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡①✐st❡ g(s)✱ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ n✱ ❞❡ ♠♦❞♦ q✉❡ t♦❞♦ ✐♥t❡✐r♦n ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛

❞❡ ♥♦ ♠á①✐♠♦ g(s) s✲és✐♠❛s ♣♦tê♥❝✐❛s✳ ❈♦♠♦ ❢♦✐ ❞✐t♦✱ ❡❧❡ ❛♣❡♥❛s ❞❡♠♦♥str♦✉ ❛

❡①✐stê♥❝✐❛ ❞❡ g(s) ♥ã♦ ❡①♣❧✐❝✐t♦✉ ♥❡♥❤✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ♠❡s♠♦✳

■r❡♠♦s ❡st✉❞❛r r❡s✉❧t❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s q✉❡ ♣♦ss✉❡♠ r❡♣r❡s❡♥t❛çã♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❞❡♠♦s♥tr❛r❡♠♦s ♦ t❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡ ♦ q✉❛❧ ❝❛r❛❝t❡r✐③❛ ♦s ✐♥t❡✐r♦s q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ s♦♠❛ ❞❡ q✉❛tr♦ q✉❛❞r❛❞♦s ❡ ❢❛❧❛r❡♠♦s ✉♠ ♣♦✉❝♦ s♦❜r❡ ♦ r❡s✉❧t❛❞♦ ❞❡ ❊✉❧❡r ♦ q✉❛❧ ❝❛r❛❝t❡r✐③❛ ♦s ♣r✐♠♦s q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❛❧é♠ ❞❡ ❡st✉❞❛r♠♦s r❡s✉❧t❛❞♦s q✉❡ ♠♦str❛♠ q✉❛♥❞♦ ✉♠ ♥ú♠❡r♦ ♥ã♦ é ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ três q✉❛❞r❛❞♦s ❝❤❡❣❛♥❞♦ ❛ ❢❛❧❛r ✉♠ ♣♦✉❝♦ s♦❜r❡ ❛ té❝♥✐❝❛ ❞♦ ❞❡s❝❡♥s♦ ✐♥✜♥✐t♦ ❞❡ ❋❡r♠❛t ❡ ❢❛③❡♥❞♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳

✷✳✷ ❙♦♠❛ ❞❡ ❞♦✐s ◗✉❛❞r❛❞♦s

■r❡♠♦s ❡st✉❞❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ♥♦s ♣❡r♠✐t✐rã♦ ❝❛r❛❝t❡r✐③❛r t♦❞♦s ♦s ✐♥t❡✐r♦s q✉❡ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♦✉ s❡❥❛✱ t♦❞♦s ♦s ✈❛❧♦r❡s ✐♥t❡✐r♦s ❞❡ n ❞❡ ♠♦❞♦ q✉❡

x2+y2 =n ✭✷✳✶✮

❛♣r❡s❡♥t❛ s♦❧✉çã♦ ❡♠ ✐♥t❡✐r♦s✳ ▼♦str❛r❡♠♦s ❛ s❡❣✉✐r ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❣❛r❛♥t❡ ♦ s❡❣✉✐♥t❡✿ s❡ ❞♦✐s ♥ú♠❡r♦s ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❡❧❡s t❛♠❜é♠ ♦ ♣♦❞❡✳

✷✳✷✳ ❙❖▼❆ ❉❊ ❉❖■❙ ◗❯❆❉❘❆❉❖❙

▲❡♠❛ ✷✳✶ ❙❡ u ❡ v sã♦ ❝❛❞❛ ✉♠ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ uv

t❛♠❜é♠ é✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦u ❡v ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛✲

❞r❛❞♦s ❡♥tã♦ ❡①✐st❡♠ a✱b✱c ❡d ✐♥t❡✐r♦s t❛✐s q✉❡u=a2

+b2 _❡

v =c2

+d2_{✱ ❞❡✈❡♠♦s}

♠♦str❛r q✉❡uv t❛♠❜é♠ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♦✉

s❡❥❛✱ q✉❡ ❡①✐st❡♠ s ❡t ✐♥t❡✐r♦s t❛✐s q✉❡uv =s2

+t2_{✳ ❉❛í✱}

uv = (a2

+b2

)(c2

+d2

) = a2

c2

+a2

d2

+b2

c2

+b2

d2

=a2

c2

+b2

d2

+a2

d2

+b2

c2

.

❆❣♦r❛ ✈❛♠♦s s♦♠❛r ❡ s✉❜tr❛✐r2(ad)(bc)✳ ❖❜t❡♥❞♦✱

uv = (a2

+b2

)(c2

+d2

) =a2

c2

+b2

d2

+a2

d2

+b2

c2

+ 2(ac)(bd)₋2(ac)(bd)

❡ ✜♥❛❧♠❡♥t❡ t❡♠♦s

uv = (ac)2

+ 2(ac)(bd) + (bd)2

+ (ad)2

−2(ad)(bc) + (bc)2

= (ac+bd)2

+ (ad₋bc)2

.

❊♥❝♦♥tr❛♠♦s s❡ t ❞❡ ♠♦❞♦ q✉❡uv =s2

+t2✱ q✉❡ é ❥✉st❛♠❡♥t❡ ♦ q✉❡ q✉❡rí❛♠♦s

♣r♦✈❛r✳

❖ t❡♦r❡♠❛ ❛❜❛✐①♦ ♥♦s ❢♦r♥❡❝❡ ❝♦♥❞✐çõ❡s ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♣r✐♠♦s q✉❡ s❡ r❡♣r❡s❡♥✲ t❛♠ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

❚❡♦r❡♠❛ ✷✳✶ ❙❡♥❞♦ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❛ ❡q✉❛çã♦ x2

+y2

= p ♣♦ss✉✐ s♦❧✉çã♦

✐♥t❡✐r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p= 2 ♦✉ p_≡1 (mod 4)✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❞♦ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡p= 2♦✉p_≡1 (mod 4)✱ ❞❡✈❡♠♦s

♠♦str❛r q✉❡ ❛ ❡q✉❛çã♦ x2

+y2

=p✱ ♦♥❞❡ p é ♣r✐♠♦✱ ♣♦ss✉✐ s♦❧✉çã♦ ✐♥t❡✐r❛✳

❉❡ ❢❛t♦✱ s❡ x = 1 ❡ y = 1 t❡♠♦s p = 2 = 12

+ 12_{✱ ❛ss✐♠}

p = 2 r❡s♦❧✈❡ ♦ ♥♦ss♦

♣r♦❜❧❡♠❛✳ ❇❛st❛ ♠♦str❛r q✉❡ p _≡ 1 (mod 4) t❡♠ q✉❡ ♦❝♦rr❡r✳ ❙❛❜❡♠♦s q✉❡ ♣❛r❛

t♦❞♦ ♣r✐♠♦ í♠♣❛r p✱ p _≡ 1 (mod 4) ♦✉ p _≡ 3 (mod 4)✳ ▲❡♠❜r❡♠♦s ❞♦ s❡❣✉✐♥t❡

❢❛t♦✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ a✱ a2

≡0 (mod 4)♦✉a2

≡1 (mod 4)✱ ❡st❡ ❢❛t♦ é ❢á❝✐❧ ❞❡ s❡r

♠♦str❛❞♦✱ s❡♥❞♦ a ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✱ s❛❜❡♠♦s q✉❡ ♦s ♣♦ssí✈❡✐s r❡st♦s ❞❛ ❞✐✈✐sã♦

❞❡ a ♣♦r q✉❛tr♦ sã♦✱ 0✱ 1✱ 2 ❡ 3✳ ❉❛í✱ a _≡ 0,1,2, ♦✉ 3 (mod 4)✱ ❛ss✐♠✱ a _≡ 0 (mod 4) ♦♥❞❡ ♦❜t❡♠♦sa2

≡02

= 0 (mod 4)✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ s❡♥❞♦a_≡1 (mod 4)

t❡r❡♠♦s a2

≡ 12

= 1 (mod 4)✱ a _≡ 2 (mod 4) ❡♥tã♦ a2

≡ 22

= 4 _≡ 0 (mod 4)

❡ ✜♥❛❧♠❡♥t❡✱a _≡ 3 (mod 4) ❡♥tã♦ a2

≡ 32

= 9 _≡ 1 (mod 4)✱ ♣♦rt❛♥t♦ t❡♠♦s q✉❡

a2

≡0♦✉1 (mod 4)✳ ❙❛❜❡♥❞♦ q✉❡a2

≡0 (mod 4)♦✉a2

≡1 (mod 4)❡x2

+y2

=p

♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ p _≡ 1 (mod 4)✱ ❞❡ ❢❛t♦❀ ♦ q✉❡ ❞❡✈❡♠♦s ♠♦str❛r é q✉❡ ❛

❝♦♥❣rê♥❝✐❛ p _≡ 3 (mod 4) s❡♥❞♦ p ♣r✐♠♦ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡ ❛❝♦♥t❡❝❡r✱ s✉♣♦♥❞♦✱ x2

≡ y2

≡ 0 (mod 4) t❡r❡♠♦s x2

+y2

≡ 0 + 0 (mod 4) ❧♦❣♦ p _≡ 0 (mod 4)✱ ❞❛

♠❡s♠❛ ❢♦r♠❛ s❡ x2

≡y2

≡1 (mod 4)❡♥tã♦ x2

+y2

≡1 + 1 (mod 4) t❡r❡♠♦sp_≡2 (mod 4) ❡ ✜♥❛❧♠❡♥t❡ s❡ x2

≡ 0 (mod 4) ❡ y2

≡ 1 (mod 4)✱ ❛ss✐♠ x2

+y2

≡ 0 + 1 (mod 4) ♦❜t❡♠♦sp_≡1 (mod 4)✳ P♦rt❛♥t♦✱ ❛ ú♥✐❝❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦ssí✈❡❧ ❞❡ ♦❝♦rr❡r

é p_≡1 (mod 4)✳

❙✉♣♦♥❞♦ q✉❡p= 2♦✉p_≡1 (mod 4)♠♦str❛r❡♠♦s q✉❡ t♦❞♦ps❛t✐s❢❛③❡♥❞♦p_≡1 (mod 4) ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳ ▲❡♠❜r❡ q✉❡ ♣❛r❛ p= 2 ❥á

s❛❜❡♠♦s q✉❡ ❡st❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ 2 = 12

+ 12✳

❚♦♠❡♠♦s ❛❣♦r❛ ✉♠ ♣r✐♠♦pq✉❡ s❛t✐s❢❛③p_≡1 (mod 4)❡ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ 1.3✱

♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❡①✐st❡ x ✐♥t❡✐r♦✱ t❛❧ q✉❡ x2

≡ −1 (mod p)✳ ❱❛♠♦s ❞❡✜♥✐r ❛

s❡❣✉✐♥t❡ ❢✉♥çã♦ f(u, v) = u+xv ❡ ❝♦♥s✐❞❡r❡♠♦s m = [√p]✳ ❙❛❜❡♥❞♦ q✉❡ √p ♥ã♦

é ✉♠ ✐♥t❡✐r♦✱ t❡♠♦s q✉❡ m < √p < m+ 1 ✳ ❚♦♠❡♠♦s ♦s ♣❛r❡s (u, v) ❞❡ ✐♥t❡✐r♦s

♦♥❞❡ 0 _≤ u _≤ m ❡ 0 _≤ v _≤ m✱ ♦♥❞❡ ♦❜s❡r✈❛♥❞♦ ♦s ✐♥t❡r✈❛❧♦s ❝♦♥❝❧✉í♠♦s q✉❡ u

✷✳✷✳ ❙❖▼❆ ❉❊ ❉❖■❙ ◗❯❆❉❘❆❉❖❙

♣♦❞❡ ❛ss✉♠✐r m+ 1 ✈❛❧♦r❡s ❡ v t❛♠❜é♠✳ ❉❛í ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s

(u, v) é(m+ 1)2✳ ❈♦♠♦

m+ 1>√pt❡♠♦s q✉❡ (m+ 1)2

>(√p)2✱ ❞á✐ ♦❜t❡♠♦s q✉❡

(m+1)2

> p✱ ❛ss✐♠ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣❛r❡s é s✉♣❡r✐♦r ❛p✳ ❙❛❜❡♠♦s q✉❡ ✉♠ s✐st❡♠❛

❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦pt❡♠ ❡①❛t❛♠❡♥t❡p❡❧❡♠❡♥t♦s✱ s❡ ❝♦♥s✐❞❡r❛r♠♦sf(u, v)

♠ó❞✉❧♦ p t❡r❡♠♦s ♠❛✐s ♥ú♠❡r♦s ❞♦ q✉❡ ❝❧❛ss❡s ❞❡ r❡sí❞✉♦s✱ ❞❛í ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛

❝❛s❛ ❞♦s ♣♦♠❜♦s ❡①✐st❡♠ ♣❡❧♦ ♠❡♠♦s ❞♦✐s ♣❛r❡s ❞✐st✐♥t♦s (u1, v1) ❡ (u2, v2) ❝♦♠

❝♦♦r❞❡♥❛❞❛s s❛t✐s❢❛③❡♥❞♦ 0 _≤ ui ≤ m ❡ 0 ≤ vi ≤ m ♦♥❞❡ (i = 1,2)✱ ♣❛r❛ ♦s

q✉❛✐s f(u1, v1) ≡ r (mod p) ❡ f(u2, v2) ≡ r (mod p)✱ ♦✉ s❡❥❛✱ f(u1, v1) ≡ f(u2, v2)

(mod p)✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ u1+xv1 ≡u2+xv2 (mod p)✱ ✐st♦ é✱

u1+xv1−u2 ≡u2+xv2−u2 (mod p)

❡ ❛ss✐♠ ✜❝❛♠♦s ❝♦♠

u1+xv1−u2 ≡xv2 (mod p)✱

❞❛í

u1+xv1−u2−xv1 ≡xv2−xv1 (mod p)✱

♦ q✉❡ r❡s✉❧t❛ ❡♠

u1−u2 ≡xv2−xv1 (mod p)

❧♦❣♦

u1−u2 ≡ −x(v2−v1) (mod p)

❡❧❡✈❛♥❞♦ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❛❝✐♠❛ ❛♦ q✉❛❞r❛❞♦ ♦❜t❡♠♦s

(u1 −u2) 2

≡(₋x)2

(v2 −v1) 2

≡x2

(v2−v1) 2

(mod p), ✭✷✳✷✮

♣♦rt❛♥t♦✱(u1−u2)2 ≡ −1(v2−v1)2 (mod p)✱ ♣♦✐sx2 ≡ −1 (mod p)✳ ❈❤❛♠❛♥❞♦

a =u1−u2 ❡ b =v1−v2✱ t❡r❡♠♦s a2 ≡ −b2 (mod p) ❛❞✐❝✐♦♥❛♥❞♦b2 ❛ ❝♦♥❣r✉ê♥❝✐❛

t❡r❡♠♦s a2

+b2

≡ −b2

+b2

(mod p) ♦ q✉❡ r❡s✉❧t❛ ❡♠ a2

+b2

≡ 0 (mod p)✱ ❛ss✐♠

❝♦♥❝❧✉í♠♦s q✉❡ p/a2

+b2✳ ❈♦♠♦ ♦s ♣❛r❡s

(u1, v1)❡ (u2, v2) sã♦ ❞✐st✐♥t♦s ❡♥tã♦a ❡ b

♥ã♦ sã♦ ❛♠❜♦s ♥✉❧♦s✱ ✐st♦ é✱ a2

+b2

>0✳ ❙❡♥❞♦ u1 ❡ u2 ✐♥t❡✐r♦s ❞♦ ✐♥t❡r✈❛❧♦[0, m]

t❡♠♦s q✉❡ a = u1−u2 ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ −m ≤ a ≤ m✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ b =

v1−v2 ❡−m≤b ≤m✳ ❈♦♠♦m <√p❝♦♥❝❧✉í♠♦s q✉❡|a| ≤m <√p✱ ❛♥❛❧♦❣❛♠❡♥t❡

|b_{| ≤} m < √p✳ ❉❛í _|a_|2

< (√p)2

= p ❞❛ ♠❡s♠❛ ❢♦r♠❛ _|b_|2

< (√p)2

= p✱ ❛ss✐♠ a2

+b2

< p+p= 2p✳ ❈♦♠♦ p/a2

+b2 _❡

0< a2

+b2

< 2p✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ú♥✐❝♦

♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞❡ p♥❡st❡ ✐♥t❡r✈❛❧♦ é ❡❧❡ ♠❡s♠♦✱ ❞❛í a2

+b2

=p✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ r❡s✉❧t❛❞♦ ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ♦ ❛♥t❡r✐♦r ❡ ♥♦s ♣❡r♠✐t❡ ✐❞❡♥✲ t✐✜❝❛r ✐♥t❡✐r♦s q✉❡ ♣♦❞❡♠ t❡r r❡♣r❡s❡♥t❛çã♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

❚❡♦r❡♠❛ ✷✳✷ ❯♠ ✐♥t❡✐r♦n ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s s❡✱

❡ s♦♠❡♥t❡ s❡✱ t✐✈❡r ❢❛t♦r❛çã♦ ❞❛ ❢♦r♠❛✳

n = 2αpα1

1 p α2

2 ...p αr

r q β1

1 q β2

2 ...q βs

s

♦♥❞❡ pi ≡ 1 (mod 4) ❡ qj ≡ 3 (mod 4)✱ i = 1, 2, ..., r, j = 1, 2, ..., s ❡ t♦❞♦s

♦s ❡①♣♦❡♥t❡s βj sã♦ ♣❛r❡s✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❞♦ q✉❡ n t❡♠ ❢❛t♦r❛çã♦ n = 2α_pα1

1 p α2

2 ...pαrrq β1

1 q β2

2 ...qβss✱

❞❡✈❡♠♦s ♠♦str❛r q✉❡ n ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♦✉

s❡❥❛✱ ❞❡✈❡♠♦s t❡♥t❛r ❡s❝r❡✈❡r ❝❛❞❛ ❢❛t♦r ❞❡ n ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

❖❜s❡r✈❡♠♦s q✉❡ ♦ ♣r✐♠♦ 2 = 12

+ 12✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡

2α _{t❛♠❜é♠ ♣♦❞❡ s❡r}

r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ s❛❜❡♠♦s ❞♦ t❡♦r❡♠❛2.1q✉❡ t♦❞♦s

♦s pi ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❛ss✐♠✱ ♦s pαii ♣♦❞❡♠

✷✳✷✳ ❙❖▼❆ ❉❊ ❉❖■❙ ◗❯❆❉❘❆❉❖❙

s❡r r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ pα1

1 p

α2

2 ...pαrr

t❛♠❜é♠✳ ❇❛st❛ ♠♦str❛r♠♦s q✉❡ ♦s qβj

j ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠❛ s♦♠❛ ❞❡

❞♦✐s q✉❛❞r❛❞♦s✳ ❚❡♠♦s ♣♦r ❤✐♣ót❡s❡ q✉❡ t♦❞♦s ♦s βi sã♦ ♣❛r❡s✱ ♦✉ s❡❥❛✱ ❡①✐st❡β

′

i t❛❧

q✉❡ βi = 2β

′

i✱ ❧♦❣♦ q βj

j = (qj)2β

′ i = (q2

j)β

′

i✳ ◆♦t❡ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q2

j = q 2 j + 0

2✱

♦✉ s❡❥❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rq2

j ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❞❛í ❞❡ ❢♦r♠❛ ❛♥ó❧♦❣❛ ♦s

qβj

j ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♣♦rt❛♥❞♦ ✉s❛♥❞♦ ♦ ❧❡♠❛ 2.1

♥♦ ♣r♦❞✉t♦ 2α_pα1

1 p α2

2 ...pαrrq β1

1 q β2

2 ...qsβs✱ ❝♦♥❝❧✉í♠♦s q✉❡ n ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛

❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

❆❣♦r❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ n ♣♦ss❛ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s

❡ q✉❡ ❡①✐st❡ ✉♠ βj q✉❡ s❡❥❛ í♠♣❛r✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛rβ1

❝♦♠♦ s❡♥❞♦ t❛❧ í♠♣❛r✳ ❈♦♥s✐❞❡r❡♠♦s q✉❡d = (a, b)♦♥❞❡a ❡b s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ a2

+b2

= n✳ ❙❡♥❞♦ d = (a, b) ❡♥tã♦ d _| a ❡ d _| b✱ ❛ss✐♠✱ ❡①✐st❡♠ k1 ❡ k2 t❛✐s q✉❡

a=k1d ❡ b=k2d✳ ❖❜s❡r✈❡♠♦s q✉❡

a d, b d = 1

d(a, b) =

1

dd= 1✱

❧♦❣♦✱ a d, b d = k1d

d , k2d

d

= (k1, k2) = 1✳

P♦❞❡♠♦s ❛✜r♠❛r q✉❡d2

|n✱ ❞❡ ❢❛t♦✱ s❛❜❡♥❞♦ q✉❡d_|a❡d_|b❡♥tã♦a=k1d❡b=k2d

❡ a ❡ b s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ a2

+b2

=n✱ ❧♦❣♦ n = (k1d)

2

+ (k2d) 2

=k2 1d

2

+k2 2d

2

=d2

(k2 1+k

2 2)

=kd2

,

❞❛í ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ d2

|n ❡ ❛❧é♠ ❞✐ss♦ s❡ ❞✐✈✐❞✐r♠♦s ❛♠❜♦s ♦s ❧❛❞♦s ❞❛

✐❣✉❛❧❞❛❞❡ ♣♦r d2 _{♦❜t❡♠♦s}

k2 1d

2

d2 +

k2 2d

2

d2 =

kd2

d2 ✳

♦ q✉❡ r❡s✉❧t❛ ❡♠

k =k2 1 +k

2 2

❙❡♥❞♦β1 í♠♣❛r ❡ t❡♥❞♦ n =kd2 ♦♥❞❡ k =

n

d2✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ❡①♣♦❡♥t❡ ❞❡ q1

❡♠ k ❞❡✈❡ s❡r í♠♣❛r✱ ♣♦✐s ♦s ♥ú♠❡r♦sk ❡ n

d2 tê♠ ❛ ♠❡s♠❛ ❞❡❝♦♠♣♦s✐çã♦ ♣r✐♠ár✐❛✳

❈♦♠♦ ♦ ❡①♣♦❡♥t❡ ❞❡ q1 é í♠♣❛r✱ ❡♥tã♦ ❡①✐st❡ s ✐♥t❡✐r♦ t❛❧ q✉❡ k = q 2s+1

1 γ ❡ ❛ss✐♠

♣♦❞❡♠♦s ❡s❝r❡✈❡r k = q2s 1 q

1

1γ = q1q12sγ✱ ♦✉ s❡❥❛✱ q1|k ❡ s❛❜❡♥❞♦ q✉❡ (k1, k2) = 1

♣♦❞❡♠♦s ♦❜s❡r✈❛r (q1, k1) = (q1, k2) = 1✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ (q1, k1) = 1✱ t❡♠♦s

♦s s❡❣✉✐♥t❡s ❞❛❞♦s (k1, k2) = 1 ❡ q1|k✱ ❞❡ (k1, k2) = 1 ❣❛r❛♥t✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ x

❡ y t❛✐s q✉❡ xk1 +yk2 = 1✱ ❡❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡st❛ ✐❣✉❛❧❞❛❞❡ ❛♦ q✉❛❞r❛❞♦✱

♦❜t❡♠♦s

(xk1+yk2) 2

= (xk1) 2

+ 2(xk1)(yk2) + (yk2) 2

=x2

k2

1 + 2xk1yk2+y 2

k2 2

= 1.

●✉❛r❞❡♠♦s ❡st❛ ✐♥❢♦r♠❛çã♦ ♣♦r ❡♥q✉❛♥t♦✱ t❡♠♦s ❛✐♥❞❛ q✉❡q1|k✱ ♦✉ s❡❥❛✱ ❡①✐st❡

s ✐♥t❡✐r♦ ❞❡ ♠♦❞♦ q✉❡ k =q1s✱ ♠❛s ♣♦r ♦✉tr♦ ❧❛❞♦k =k12+k 2

2✱ ❧♦❣♦✱ k 2 1 +k

2 2 =q1s

❡ ❛ss✐♠ s❡❣✉❡ q✉❡ k2

2 =q1s−k12✱ ❧❡♠❜r❡♠♦s t❛♠❜é♠ q✉❡ b = k2d✱ ♦♥❞❡ d = (a, b)✱

♣♦r ✐ss♦✱ k1 =

b

d ❛❣♦r❛ ✈❛♠♦s s✉❜st✐t✉✐r ❡st❡s ✈❛❧♦r❡s ❡♠ x

2

k2

1 + 2xk1yk2+y2k22 = 1

❡ ♦❜t❡r❡♠♦s q✉❡

x2

k2

1 + 2xk1yk2+y 2

k2

2 =x

2

k2

1+ 2xk1y

b d

+y2

(q1s−k 2 1)

=x2

k2

1+ 2xk1y

b d

+y2

q1s−y 2

k2 1

= 1,

✷✳✷✳ ❙❖▼❆ ❉❊ ❉❖■❙ ◗❯❆❉❘❆❉❖❙

✈❛♠♦s ❥✉♥t❛r ♦s t❡r♠♦s q✉❡ ❝♦♥té♠k1 ❡ ♦s q✉❡ ❝♦♥té♠ q1✱ ❛ss✐♠ ✜❝❛r❡♠♦s ❝♦♠

x2

k2

1+ 2xk1y _db

−y2

k2 1 +y

2

q1s= 1✱ ✈❛♠♦s ♣♦r ❡♠ ❡✈✐❞ê♥❝✐❛ ♥❛ ❡①♣r❡ssã♦k1 ❡q1✱

❞❛í

x2

k1+ 2xy

b d

−y2

k1

k1+ (y 2

s)q1 = 1, ✭✷✳✸✮

♦❜s❡r✈❡♠♦s q✉❡t=x2

k1+2xy _db

−y2

k1❡u=y2ssã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♣♦rt❛♥t♦

❛ ❡①♣r❡ssã♦ tk1+uq1 = 1 ♥♦ ❞✐③ q✉❡q1 ❡k1 sã♦ ♣r✐♥♦s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱(q1, k1) = 1✱

❛♥❛❧♦❣❛♠❡♥t❡ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ (q1, k2) = 1✳

❯s❛♥❞♦ ❛ ♣r♦♣♦s✐çã♦1.1✱ ❣❛r❛♥t✐♠♦s q✉❡ ❡①✐st❡x❞❡ ♠♦❞♦ q✉❡k1x≡k2 modq1

❡ ❝♦♠♦ q1 |k✱ ♣♦rt❛♥t♦ k ≡0 mod q1✱ ♠❛s ❧❡♠❜r❡♠♦s q✉❡ k =k21+k2✱ ❡♥tã♦

k2 1 +k

2

2 ≡k

2 1 +k

2 2−k

2

2 ≡0−k

2 2 ≡ −k

2

2 mod q1✳

❈♦♠♦ k1x ≡k2 modq1✱ t❡♠♦s q✉❡ ❡❧❡✈❡♥❛❞♦ ❛♦ q✉❛❞r❛❞♦ ❡st❛ ❝♦♥❣rê♥❝✐❛ ♦❜✲

t❡♠♦s k2 1x

2

≡ k2

2 mod q1✳ ❆❣♦r❛ s♦♠❛♥❞♦ ❛s ❝♦♥❣r✉ê♥❝✐❛s k12 ≡ −k 2

2 mod q1 ❡

k2 1x

2

≡k2

2 mod q1✱ ✜❝❛♠♦s ❝♦♠

k2 1x

2

+k2

1 =k

2 1(x

2

+ 1) _{≡ −}k2 2+k

2

2 ≡0 mod q1✳

❋❛ç❛♠♦s ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✱ q1 ∤ k 2

1✱ ❞❡ ❢❛t♦✱ s❡♥❞♦ (q1, k2) = 1✱ t❡♠♦s q✉❡

q1 ∤k1✱ ♣♦rt❛♥t♦ ♥ã♦ ❞✐✈✐❞❡ k21✳

❱❛♠♦s ♠♦str❛r ❡st❡ ❢❛t♦✱ ♣❛r❛ ✐ss♦ ✉s❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ♣❡❧❛ ❝♦♥tr❛♣♦s✐t✐✈❛✱ ♦✉ s❡❥❛✱ s✉♣♦♥❤❛♠♦s q✉❡ q1 | k12✱ ❞á✐ q1 | k1k1✱ ❝♦♠♦ q1 é ♣r✐♠♦ ❡♥tã♦ q1 | k1 ♦✉

q1 | k1✱ ♣♦rt❛♥t♦ q1 | k1 ❡ ❛ss✐♠✱ ♠♦str❛♠♦s q✉❡ q1 ∤ k 2

1✳ ❈♦♠♦ q1 é ♣r✐♠♦ ❡

q1 |k21(x 2

+ 1) ❡♥tã♦ q1 | k12 ♦✉ q1 | (x2+ 1)✱ ♠❛sq12 ∤ k 2

1 ♣♦rt❛♥t♦✱ q1 | (x2+ 1)✱ ♦✉

s❡❥❛✱ x2

≡ −1 mod q1✳ ❖❜s❡r✈❡♠♦s q✉❡ ❛ ❡q✉❛çã♦x2 ≡ −1 mod q1 ♣♦ss✉✐ s♦❧✉çã♦

♣❛r❛ q1 ≡ 3 (mod 4) ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ♣r♦♣♦s✐çã♦ 1.1✱ ♣♦rt❛♥t♦ t♦❞♦s ♦s β

′

js sã♦

♣❛r❡s✳

✷✳✸ ❙♦♠❛ ❞❡ ❚rês ◗✉❛❞r❛❞♦s

❖ q✉❡ ❢❛r❡♠♦s ♥❡st❛ s❡çã♦ é ❡①✐❜✐r ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ♥ú♠❡r♦s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ três q✉❛❞r❛❞♦s✳

❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ q✉❡ s❡ s❡❣✉❡ ♥♦s ❞✐③ q✉❡ t♦❞♦ ✐♥t❡✐r♦ q✉❡ ❞❡✐①❛ r❡st♦ 7

q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 8♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ três q✉❛❞r❛❞♦s✳

❚❡♦r❡♠❛ ✷✳✸ ❚♦❞♦ ✐♥t❡✐r♦ ❞❛ ❢♦r♠❛ 8a+ 7 ❝♦♠ a _∈ Z ♥ã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦

❝♦♠♦ ❛ s♦♠❛ ❞❡ três q✉❛❞r❛❞♦s✳

❉❡♠♦♥str❛çã♦✿ ❚♦♠❡♠♦sn ✐♥t❡✐r♦✳ ❙❛❜❡♠♦s q✉❡ ❛♦ ❞✐✈✐❞✐r♠♦sn ♣♦r8♣♦❞❡✲

♠♦s ♦❜t❡r ❝♦♠♦ r❡st♦ ❛❧❣✉♠ ❞♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s0✱ 1✱ 2✱ 3✱ 4✱5✱6♦✉7✱ ♣♦rt❛♥t♦✱

a _≡ 0 (mod 8) ♦✉ a _≡ 1 (mod 8)✱ a _≡ 2 (mod 8)✱ a _≡ 3 (mod 8)✱ a _≡4 (mod 8)✱

a_≡5 (mod 8)✱ a_≡6 (mod 8)✱ a_≡7 (mod 8)✳

❉❛í✱

a2

≡02

= 0 (mod 8)

a2

≡12

= 1 (mod 8)

a2

≡22

= 4 (mod 8)

a2

≡32

= 9_≡1 (mod 8)

a2

≡42

= 16_≡0 (mod 8)

a2

≡52

= 25_≡1 (mod 8)

a2

≡62

= 36_≡4 (mod 8)

a2

≡72

= 49_≡1 (mod 8).

❈♦♥❝❧✉í♠♦s ❛ss✐♠✱ q✉❡ a2

≡ 0✱ 1 ♦✉ 4 (mod 8)✳ ❆❣♦r❛✱ ♦❜s❡r✈❡♠♦s q✉❡ r❡❛❧✐✲

③❛♥❞♦ t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s ♣❛r❛ ❛s s♦♠❛s ❞♦s q✉❛❞r❛❞♦s ♥ã♦ é ♣♦ssí✈❡❧ ♦❜t❡r a2

+b2

+c2

≡ 7 (mod 8)✳ ❉❡ ❢❛t♦✱ ✈❛♠♦s ❞❡s❝r❡✈❡r t♦❞❛s ❛s ♣♦sss✐❜✐❧✐❞❛❞❡s

♣❛r❛ ❛ s♦♠❛ a2

+b2

+c2_✳