NÚMEROS PRIMOS COMO SOMA DE DOIS QUADRADOS

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❉❖◆■❩❊❚❊ ❘❖❈❍❆ ❉❊ ❇❘■❚❚❊❙

◆Ú▼❊❘❖❙ P❘■▼❖❙ ❈❖▼❖ ❙❖▼❆ ❉❊ ❉❖■❙

◗❯❆❉❘❆❉❖❙

❈❆▼P❖ ●❘❆◆❉❊

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❉❖◆■❩❊❚❊ ❘❖❈❍❆ ❉❊ ❇❘■❚❚❊❙

◆Ú▼❊❘❖❙ P❘■▼❖❙ ❈❖▼❖ ❙❖▼❆ ❉❊ ❉❖■❙

◗❯❆❉❘❆❉❖❙

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ❊▲■❙❆❇❊❚❊ ❙❖❯❙❆ ❋❘❊■❚❆❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✕ ■◆▼❆✴❯❋▼❙✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡✳

❈❆▼P❖ ●❘❆◆❉❊

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◆Ú▼❊❘❖❙ P❘■▼❖❙ ❈❖▼❖ ❙❖▼❆ ❉❊ ❉❖■❙

◗❯❆❉❘❆❉❖❙

❉❖◆■❩❊❚❊ ❘❖❈❍❆ ❉❊ ❇❘■❚❚❊❙

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐✲ s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

❆♣r♦✈❛❞♦ ♣❡❧❛ ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s ✲ ❯❋▼❙ Pr♦❢✳ ❉r✳ ❈❧❛✉❞❡♠✐r ❆♥✐③ ✲ ❯❋▼❙

Pr♦❢✳ ❉r✳ ▲✐♥♦ ❙❛♥❛❜r✐❛ ✲ ❯❋●❉

❈❆▼P❖ ●❘❆◆❉❊

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❊♣í❣r❛❢❡

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❆●❘❆❉❊❈■▼❊◆❚❖❙

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦✲ ♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛ ❡ ♠❡ ❛❥✉❞❛r ❛ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s q✉❡ ❝♦♠ s✉❛ ✐♠❡♥s❛ s❛❜❡❞♦r✐❛ ❡ ♣❛❝✐ê♥❝✐❛ ♠❡ ❣✉✐♦✉ ♠✉✐t♦ ❜❡♠ ❞✉r❛♥t❡ ♦ tr❛✲ ❜❛❧❤♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ♣♦r ♥✉♥❝❛ ♠❡❞✐r❡♠ ❡s❢♦rç♦s ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ ❛s ♠❡❧❤♦r❡s ❝♦♥❞✐çõ❡s ❞❡ ❡st✉❞♦✳

❆❣r❛❞❡ç♦ t❛♠❜é♠✱ t♦❞♦s ♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ ❞❡s❞❡ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❛té ♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞♦ ❝✉rs♦ ❞❡ ❧✐❝❡♥❝✐❛t✉r❛ ❡♠ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋▼❙✱ ♣r♦✜ss✐♦♥❛✐s ❢❛♥tást✐❝♦s q✉❡ ♠✉❞❛r❛♠ ❛ ♠✐♥❤❛ ✈✐❞❛✳

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❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡st❛❜❡❧❡❝❡r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p ♣♦ss❛ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛r✐t♠ét✐❝♦ ❝♦♠♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛❧❣é❜r✐❝♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♦♥❞❡ ❛❞♠✐t✐r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜❡♠ ❝♦♥❤❡❝✐❞♦s✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛❧❣é❜r✐❝♦ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❡ ❡♠ ♣❛rt✐❝✉❧❛r ♦ ❞♦♠í♥✐♦ ❊✉❝❧✐❞✐❛♥♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✳

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❆❜str❛❝t

❚❤✐s ✇♦r❦ ❛✐♠s t♦ ❡st❛❜❧✐s❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ♣r✐♠❡ ♥✉♠❜❡r p ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ s✉♠ ♦❢ t✇♦ sq✉❛r❡s ❢r♦♠ t✇♦ ♣♦✐♥ts ♦❢ ✈✐❡✇✿ t❤❡ ❛r✐t❤♠❡t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇ ❛♥❞ ❢r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❋✐rst✱ ✇❡ ✇✐❧❧ ✇♦r❦ ✇✐t❤ t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇❤✐❝❤ ❛❞♠✐t s♦♠❡ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✳ ❋r♦♠ t❤❡ ❛❧❣❡❜r❛✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇ ✇❡ ✇✐❧❧ st✉❞② s♦♠❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ❊✉❝❧✐❞❡❛♥ ❞♦♠❛✐♥ str✉❝t✉r❡s ❢♦r♠❡❞ ❜② ●❛✉ss✐❛♥ ✐♥t❡❣❡rs✳

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s s♦❜r❡ ◆ú♠❡r♦s Pr✐♠♦s ✸

✸ ❚❡r♥♦s P✐t❛❣ór✐❝♦s ❡ ♣r✐♠♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ✾

✸✳✶ ❚❡r♥♦s ♣✐t❛❣ór✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✸✳✷ Pr✐♠♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✹ ❊str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❡ ❢❛t♦r❛çã♦ ✷✸

✹✳✶ ❉❡✜♥✐çõ❡s✱ ❡①❡♠♣❧♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✹✳✷ ❖s ❆♥é✐s Zm ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹✳✸ ❖ ❛♥❡❧ ❞♦s P♦❧✐♥ô♠✐♦s ❑❬①❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✹ ❖ ❆♥❡❧ Z[i] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✺ ◆❛t✉r❛✐s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ✹✼

✺✳✶ Pr✐♠♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✿ ❝❛r❛❝t❡r✐③❛çã♦ ❡♠Z[i] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✷ ❚❡r♥♦s ♣✐t❛❣ór✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺✳✸ ◆❛t✉r❛✐s ❝♦♠♦ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◗✉❛♥❞♦ ✉♠ ♣r✐♠♦ p♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s❄ ■st♦ é✱ q✉❛♥❞♦ ❡①✐st❡♠ ✐♥t❡✐r♦s a ❡ b t❛✐s q✉❡ p = a2 ₊_b2_{❄ ❆♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✱ r❡s♣♦♥❞❡r❡♠♦s ❡st❛ ♣❡r❣✉♥t❛}

❛r✐t♠❡t✐❝❛♠❡♥t❡ ❡ ❛❧❣❡❜r✐❝❛♠❡♥t❡✳

❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♣r✐♠♦s q✉❡ ♣♦❞❡♠ ♦✉ ♥ã♦ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✿

✶✮ ❈♦♥s✐❞❡r❡ ♦s ♣r✐♠♦s 13 ❡ 17✳ ❖❜s❡r✈❡ q✉❡ 13 = 22 _{+ 3}2 _❡ _{17 = 1}2 _{+ (}₋₄₎2_✱

♣♦rt❛♥t♦ 13❡ 17♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

✷✮ ❏á ♦s ♣r✐♠♦s7❡ 11♥ã♦ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♣♦✐s

♥ã♦ ❡①✐st❡♠ ✐♥t❡✐r♦s a ❡b t❛✐s q✉❡ 7 =a2₊_b2 _♦✉ _{11 =}_a2₊_b2_✳

❈♦♠ ❡①❝❡çã♦ ❞♦ 2✱ t♦❞♦s ♦s ♣r✐♠♦s ❞❡✐①❛♠ r❡st♦ 1♦✉ 3 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦s ♣♦r 4✳

❖❜s❡r✈❛♠♦s ♥♦1➸ ❡①❡♠♣❧♦ q✉❡ ♦s ♣r✐♠♦s13❡17sã♦ t❛✐s q✉❡13 = 4.3 + 1❡17 = 4.4 + 1✱ ♦✉

s❡❥❛✱ ❛♠❜♦s ❞❡✐①❛♠ r❡st♦ 1q✉❛♥❞♦ ❞✐✈✐❞✐❞♦s ♣♦r 4✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣ós ❛❞♠✐t✐r ❝♦♥❤❡❝✐❞❛s

❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♣r♦✈❛r❡♠♦s q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣r✐♠♦s q✉❡ ❞❡✐①❛♠ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦s ♣♦r 4✱ ❡ q✉❡ t♦❞♦s ♦s ♣r✐♠♦s ❞❡st❡ t✐♣♦ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s

❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ♣r♦✈❛r❡♠♦s q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣r✐♠♦s q✉❡ ❞❡✐①❛♠ r❡st♦ 3 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦s ♣♦r 4 ❡ q✉❡ ♥❡♥❤✉♠ ❞❡❧❡s ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡

❞♦✐s q✉❛❞r❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ✈❡r❡♠♦s ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ♥❛t✉r❛✐s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s

(11)

q✉❛❞r❛❞♦s✱ ♦s t❡r♥♦s ♣✐t❛❣ór✐❝♦s✳ ❯♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦ (a, b, c) é ❢♦r♠❛❞♦ ♣♦r ♥❛t✉r❛✐s t❛✐s

q✉❡ a2₊_b2 ₌_c2_{✳ ❯s❛r❡♠♦s ♦ ♠ét♦❞♦ ❞❡ ❊✉❝❧✐❞❡s ♣❛r❛ ❡♥❝♦♥tr❛r t❡r♥♦s ♣✐t❛❣ór✐❝♦s}₍_{a, b, c}₎

t❛✐s q✉❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡a ❡ b é 1✳

◆♦ ❝❛♣ít✉❧♦ ✹✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s✱ ❝♦♠ ❡①❡♠♣❧♦s q✉❡ s❡✲ rã♦ ✉s❛❞♦s ♣♦st❡r✐♦r♠❡♥t❡✳ ◆♦ ❝❛♣ít✉❧♦ ✺✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❜✉s❝❛r❡♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ ✉♠ ♣r✐♠♦ p s❡r s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✭_Z[i]✮✳ P♦st❡r✐✲

♦r♠❡♥t❡✱ ❝❛r❛❝t❡r✐③❛r❡♠♦s ♥♦✈❛♠❡♥t❡ ♦s t❡r♥♦s ♣✐t❛❣ór✐❝♦s ✉s❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✭Z[i]✮ ❡ ❣❡♥❡r❛❧✐③❛r❡♠♦s ♦ r❡s✉❧t❛❞♦ ❡st❛❜❡❧❡❝✐❞♦ ♣❛r❛ ♥ú♠❡r♦s ♣r✐♠♦s ♣❛r❛ ✉♠

♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❛❧q✉❡r✳

(12)

❈❛♣ít✉❧♦ ✷

❘❡s✉❧t❛❞♦s ❇ás✐❝♦s s♦❜r❡ ◆ú♠❡r♦s

Pr✐♠♦s

◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆❞♠✐t✐r❡♠♦s ❛❧✲ ❣✉♥s ❢❛t♦s ❝♦♥❤❡❝✐❞♦s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ ❖ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ❡ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛ ♥ã♦ s❡rã♦ ❞❡♠♦♥str❛❞♦s✳

❚❡♦r❡♠❛ ✶✳ ✭❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✮ ❉❛❞♦s a ❡ b ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ b ₆= 0✱

❡♥tã♦ ❡①✐st❡♠ ú♥✐❝♦s q ❡ r✱ ✐♥t❡✐r♦s✱ t❛✐s q✉❡✿

a=bq+r, 0_≤r <_|b_|.

❉❛❞♦s ❞♦✐s ✐♥t❡✐r♦sa❡b✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦a|b ♣❛r❛ ✐♥❞✐❝❛r q✉❡aé ✉♠ ❞✐✈✐s♦r ❞❡b✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ct❛❧ q✉❡ b=ac ❡a ∤b ✐♥❞✐❝❛rá q✉❡ a ♥ã♦ é ❞✐✈✐s♦r ❞❡b✳

❆ ♥♦t❛çã♦mdc(a, b)✐♥❞✐❝❛rá ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ♦s ✐♥t❡✐r♦s a❡b✱ ♥ã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥✉❧♦s✳ ▲❡♠❜r❛♠♦s q✉❡✱ s❡ d = mdc(a, b) ❡♥tã♦ ❡①✐st❡♠ r ❡ s ✐♥t❡✐r♦s t❛✐s q✉❡d=ra+sb✳

(13)

❉❡✜♥✐çã♦ ✶✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡1q✉❡ só ♣♦ss✉✐ ❝♦♠♦ ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s1❡

❡❧❡ ♣ró♣r✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦✳

❙❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ♦s s❡❣✉✐♥t❡s ❢❛t♦s✿

•❙❡p ❡q sã♦ ♣r✐♠♦s t❛✐s q✉❡p|q ❡♥tã♦ p=q✳

•❙❡p é ♣r✐♠♦ ❡ p_∤a ❡♥tã♦ ♦ mdc(p, a) = 1✳

▲❡♠❛ ✶✳ ✭▲❡♠❛ ❞❡ ●❛✉ss✮ ❙❡❥❛♠ a, b❡ c ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❙❡ a_|bc ❡ mdc(a, b) = 1✱ ❡♥tã♦

a_|c✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦mdc(a, b) = 1 s❡❣✉❡ q✉❡ ❡①✐st❡♠ ✐♥t❡✐r♦sr ❡s t❛✐s q✉❡

ra+sb= 1

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ♣♦rc✱ ♦❜t❡♠♦s

rac+sbc =c

♦♥❞❡a _|rac ❡a _|sbc✱ ♣♦rt❛♥t♦ a_|c✳

Pr♦♣♦s✐çã♦ ✶✳ ✭Pr♦♣r✐❡❞❛❞❡ ❋✉♥❞❛♠❡♥t❛❧ ❞♦s ◆ú♠❡r♦s Pr✐♠♦s✮ ❙❡❥❛♠ a, b, p ✐♥t❡✐r♦s ❝♦♠ p ♣r✐♠♦✳ ❙❡ p_|ab ❡♥tã♦ p_|a ♦✉ p_|b✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡p|ab❡ q✉❡ p∤a✳ ❙❡❣✉❡ q✉❡mdc(p, a) = 1❡ ❛ss✐♠✱ ✉s❛♥❞♦

♦ ❧❡♠❛ ❞❡ ●❛✉ss✱ ❝♦♥❝❧✉í♠♦s q✉❡p|b✳

❚❡♦r❡♠❛ ✷✳ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ n ₆= 0,₋1,1✱

❡①✐st❡♠ ♣r✐♠♦s p1 < . . . < pn✱ ❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s α1, . . . , αn ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱

t❛✐s q✉❡ n=_±pα1

1 · · ·pαnn✳

▲❡♠❛ ✷✳ ❙❡❥❛ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❖s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠❜✐♥❛tór✐♦s p i

!

✱ ♦♥❞❡ 0 < i < p✱ sã♦ t♦❞♦s ❞✐✈✐sí✈❡✐s ♣♦r p✳

(14)

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ ✐♥t❡✐r♦ p i

!

=p_·(p−1)·..._i·_!(p−i+1)✳ P❛r❛i= 1 t❡♠♦s p 1

!

=p✱ ♣♦rt❛♥t♦ ♦ r❡s✉❧t❛❞♦ ✈❛❧❡ tr✐✈✐❛❧♠❡♥t❡✳ P❛r❛1< i < p✱ ✈❛❧❡ q✉❡i!|p(p−1)·. . .·(p−i+ 1)✳

❈♦♠♦mdc(i!, p) = 1 ✭♣♦✐si < p✮✱ s❡❣✉❡ ❞♦ ▲❡♠❛ ❞❡ ●❛✉ss q✉❡✱ i!|(p−1)·. . .·(p−i+ 1)✱

❛ss✐♠ p_| p i

!

✳

❚❡♦r❡♠❛ ✸✳ ✭ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p✱ t❡♠✲s❡ q✉❡✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ a✱ p ❞✐✈✐❞❡ ♦ ♥ú♠❡r♦ ap₋a✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ♦ ♣r✐♠♦ p = 2 t❡♠♦s q✉❡ 2 | a2₋_a _{✱ ♣♦✐s} _a2₋_a ₌_a₍_a₋₁₎ _{é s❡♠♣r❡}

♣❛r✳

❙✉♣♦♥❤❛♠♦sp♣r✐♠♦ í♠♣❛r✳ ◆❡ss❡ ❝❛s♦✱ ❝♦♠♦(−a)p₋₍₋_a_{) =}₋_ap₊_a₌₋₍_ap₋_a₎_✱

❜❛st❛ ♠♦str❛r ♦ r❡s✉❧t❛❞♦ ♣❛r❛ a_≥0✳ ❱❛♠♦s ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ✉s❛♥❞♦ ✐♥❞✉çã♦ s♦❜r❡a✳ ❖ r❡s✉❧t❛❞♦ ✈❛❧❡ ♣❛r❛ a= 0✱ ♣♦✐s p é ✉♠ ❞✐✈✐s♦r ❞❡0✳

❙✉♣♦♥❤❛♠♦s ♦ r❡s✉❧t❛❞♦ ✈á❧✐❞♦ ♣❛r❛ a✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ ❝♦♥t✐♥✉❛ ✈á❧✐❞♦ ♣❛r❛ a+ 1✳ ❯s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ t❡♠♦s q✉❡

(a+ 1)p₋₍_a_{+ 1) =}_ap₋_a₊ p 1

!

ap−1₊_{. . .}₊ p

p₋1

!

a

❯s❛♥❞♦ ♦ ❧❡♠❛ ❡ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡p|(a+ 1)p₋₍_a_{+ 1)}_✳

❈♦r♦❧ár✐♦ ✶✳ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ a ❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ t❛❧ q✉❡ p ∤ a✱ ❡♥tã♦

p_|ap−1₋₁_✳

❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t t❡♠♦s q✉❡ p _| a(ap−1 ₋₁₎ _{❡ ❝♦♠♦}

p∤a✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ❝♦♥❝❧✉í♠♦s q✉❡

p|ap−1₋₁

(15)

❚❡♦r❡♠❛ ✹✳ ❊①✐st❡♠ ✐♥✜♥✐t♦s ♥ú♠❡r♦s ♣r✐♠♦s✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✱ ❞✐❣❛♠♦s p1, p2, . . . pn✳ ❈♦♥s✐❞❡r❡ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧a=p1p2·. . .·pn+ 1✭♦ ♣r♦❞✉t♦ ❞❡ t♦❞♦s ♦s ♣r✐♠♦s

♠❛✐s1✮✳ P❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✱ ♦ ♥ú♠❡r♦ a ♣♦ss✉✐ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦p❡ ♣♦rt❛♥t♦p=pi✱ ❝♦♠1≤i≤n✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡p|p1p2·. . .·pn❡ ❞❛íp|1 = a−p1p2·. . .·pn

♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

❖❜s❡r✈❛çã♦ ✶✳ ❊ss❛ ❞❡♠♦♥str❛çã♦ ❞❛❞❛ ♣♦r ❊✉❝❧✐❞❡s✱ ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❞❛s ♣ér♦❧❛s ❞❛ ♠❛t❡✲ ♠át✐❝❛✱ é ♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ❞❡ ♣r♦✈❛ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳

❖❜s❡r✈❛♠♦s q✉❡ t♦❞♦ ♣r✐♠♦ í♠♣❛rpé ❞❛ ❢♦r♠❛4k+1♦✉4k+3✱ ♦✉ s❡❥❛✱ ❞✐✈✐❞✐♥❞♦

✉♠ ♣r✐♠♦ í♠♣❛r ♣♦r4 ❡♥❝♦♥tr❛r❡♠♦s r❡st♦1 ♦✉3✳

❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦r4♦❜t❡r❡♠♦s

r❡st♦s 0,1,2 ♦✉ 3✱ ❛ss✐♠ p = 4k,4k+ 1,4k+ 2 ♦✉ 4k+ 3 ❡ ❝♦♠♦ p é í♠♣❛r ❝♦♥❝❧✉í♠♦s q✉❡ p= 4k+ 1 ♦✉ 4k+ 3✳ ▼♦str❛r❡♠♦s ❛ s❡❣✉✐r q✉❡ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣r✐♠♦s ❞❛s ❞✉❛s

❢♦r♠❛s✿ 4k+ 1 ❡ 4k+ 3✳

Pr♦♣♦s✐çã♦ ✷✳ ❊①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣r✐♠♦s ❞❛ ❢♦r♠❛4k+ 3✳

❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r♦✱ ♦❜s❡r✈❡ q✉❡ ♦ ❝♦♥❥✉♥t♦A=_{4k+ 1_|kǫ_N_}é ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❉❡ ❢❛t♦✱ (4k1 + 1)(4k2+ 1) = 4(4k1k2+k1+k2) + 1 ǫ A✳

❯s❛♥❞♦ ❛ ♠❡s♠❛ ✐❞❡✐❛ ❞❡ ❊✉❝❧✐❞❡s✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❞❛ ❢♦r♠❛ 4k + 3✱ ❞✐❣❛♠♦s 3 < p2 < . . . < pn✳ ❈♦♥s✐❞❡r❡ a = 4(p2p3· · ·pn) + 3❡ ✉♠ p♣r✐♠♦ ❞✐✈✐s♦r ❞❡a✳ ✽❙❡❣✉❡ q✉❡pé ❞✐❢❡r❡♥t❡ ❞♦s ♣r✐♠♦s3, p2, . . . , pn✳

❉❡ ❢❛t♦✱ s❡ p = 3 s❡❣✉❡ q✉❡ 3| a−3 = 4(p2p3 · · ·pn)✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

❆♥❛❧♦❣❛♠❡♥t❡ s❡p=pi✱ 2≤i≤n✱ s❡❣✉❡ q✉❡pi |a−4(p2p3· · ·pn) = 3✱ ♦ q✉❡ é ♥♦✈❛♠❡♥t❡

✉♠❛ ❝♦♥tr❛❞✐çã♦✳

❆ss✐♠ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡a ❡♠ ❢❛t♦r❡s ♣r✐♠♦s só ♣♦❞❡ t❡r ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ A✱ ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❈❤❡❣❛♠♦s ❛ ✉♠ ❛❜s✉r❞♦ ♣♦✐saé ❞❛ ❢♦r♠❛4k+ 3✳

(16)

❱❛♠♦s ✉s❛r ♦ ❧❡♠❛ s❡❣✉✐♥t❡ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣r✐♠♦s ❞❛ ❢♦r♠❛4k+ 1✳

▲❡♠❛ ✸✳ ❚♦❞♦ ❞✐✈✐s♦r ♣r✐♠♦ í♠♣❛r ❞❡ x2 _{+ 1}_{✱ ❝♦♠} _x _{♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡} ₁_{✱ é ❞❛ ❢♦r♠❛}

4k+ 1✳

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡4_∤(x2_{+ 1)}_{✳ ❉❡ ❢❛t♦✱ s❡}_x_{= 2}_k_{✱ ❡♥tã♦}_x2_{+ 1 =}

(2k)2_{+ 1 = 4(}_k2_{) + 1}_{✱ ❡✱ s❡} _x_{= 2}_k_{+ 1} _❡♥tã♦ _x2_{+ 1 = (2}_k_{+ 1)}2_{+ 1 = 4(}_k2₊_k_{) + 2}_{✱ ❧♦❣♦ ♥♦s}

❞♦✐s ❝❛s♦s✱ 4∤ (x2_{+ 1)}_{✳ ❙❡❣✉❡ q✉❡} _x2_{+ 1} _{♥ã♦ é ♣♦tê♥❝✐❛ ❞❡} ₂ _{❡ ♣♦rt❛♥t♦ ♣♦ss✉✐ ✉♠ ❞✐✈✐s♦r}

♣r✐♠♦ í♠♣❛r✱ ❞✐❣❛♠♦sp✳ ❚❡♠♦s q✉❡ p−1

2 ǫ N❡✱ ♣❛r❛ ❛❧❣✉♠ t ǫ N✱

x2 =tp−1

❊❧❡✈❛♥❞♦ ❛ ♣♦tê♥❝✐❛ p−1

2 ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ❡ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛

❞♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ♦❜t❡♠♦s✿

xp−1 ₌

        

kp+ 1 _se p−1

2 ´e par

kp₋1 _se p−1

2 ´e ´ımpar

❙✉♣♦♥❤❛♠♦s xp−1 ₌ _kp₋₁_{✱ ❧♦❣♦} _xp−1 ₋_{1 =} _kp₋₂_{✳ ❈♦♠♦} _p _| _x2 _{+ 1}_{✱ s❡❣✉❡}

q✉❡ p _∤ x✳ ❆❣♦r❛ ♣❡❧♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ t❡♠♦s q✉❡ p _| xp−1 ₋₁ _{❡ ♣♦rt❛♥t♦}

p_|kp₋(xp−1₋_{1) = 2}_{✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳}

P♦rt❛♥t♦ p−1

2 t❡♠ q✉❡ s❡r ♣❛r✱ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ p= 4k+ 1✳

Pr♦♣♦s✐çã♦ ✸✳ ❊①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣r✐♠♦s ❞❛ ❢♦r♠❛4k+ 1✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❛ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣r✐♠♦s ❞❛ ❢♦r♠❛ 4k+ 1✱ ❞✐❣❛♠♦s p1, p2, . . . , pn✳ ❈♦♥s✐❞❡r❡

a = 4(p1·p2· · ·pn)2+ 1

(17)

❈♦♠♦pi ∤a✱ ♣❛r❛ t♦❞♦ i= 1, . . . , n✱ ❝❛s♦ ❝♦♥tr❛r✐♦p|1✱ ❝♦♥❝❧✉í♠♦s q✉❡ a ♣♦ss✉✐

✉♠ ❞✐✈✐s♦r ♣r✐♠♦ ❞❛ ❢♦r♠❛4k+ 3✱ ♦ q✉❡ ❝♦♥tr❛r✐❛ ♦ ❧❡♠❛✳

(18)

❈❛♣ít✉❧♦ ✸

❚❡r♥♦s P✐t❛❣ór✐❝♦s ❡ ♣r✐♠♦s ❝♦♠♦ s♦♠❛

❞❡ ❞♦✐s q✉❛❞r❛❞♦s

✸✳✶ ❚❡r♥♦s ♣✐t❛❣ór✐❝♦s

◆❡st❛ s❡çã♦ ❡st✉❞❛♠♦s ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♠ ❧❛❞♦s ✐♥t❡✐r♦s✳ ❙❡ ✐♥❞✐❝❛r♠♦s ♣♦r a✱ b ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞♦s ❝❛t❡t♦s ❡ c ❛ ♠❡❞✐❞❛ ❞❛ ❤✐♣♦t❡♥✉s❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦s ❞✐③ q✉❡a2₊_b2 ₌_c2_{✳ ❱❛❧❡ t❛♠❜é♠ ❛ r❡❝í♣r♦❝❛✱ s❡} _a✱

b ❡csã♦ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡a2₊_b2 ₌_c2 _{❡♥tã♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦ ❡}

❛ ❤✐♣♦t❡♥✉s❛ ♠❡❞❡c✳

❉❡✜♥✐çã♦ ✷✳ ❯♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦ (a, b, c) é ❢♦r♠❛❞♦ ♣♦r três ♥ú♠❡r♦s ♥❛t✉r❛✐s t❛✐s q✉❡

a2₊_b2 ₌_c2_✳

❊①❡♠♣❧♦ ✶✳ ❖s ♥ú♠❡r♦s3✱ 4✱ ❡5 ❢♦r♠❛♠ ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✱

♣♦✐s32_{+ 4}2 _{= 5}2_✳

❊①❡♠♣❧♦ ✷✳ ✳❖s ♥ú♠❡r♦s6✱ 8✱ ❡10❢♦r♠❛♠ ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✱

♣♦✐s62_{+ 8}2 _{= 10}2_✳

(19)

❖❜s❡r✈❛çã♦ ✷✳ i) ❙❡n _∈N é ✉♠ ♥ú♠❡r♦ í♠♣❛r✱ ❡♥tã♦ a=n✱b =n2₂−1 ❡ c=n2₂+1 ❢♦r♠❛♠

✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳

ii) ❙❡n _∈ N é ✉♠ ♥ú♠❡r♦ ♣❛r✱ ❡♥tã♦ a= n✱ b =

(

n₂

)

2₋1 ❡ c=

(

n₂

)

2+1 ❢♦r♠❛♠

✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳ ❉❡ ❢❛t♦✱

i)❚♦♠❛♥❞♦ n é í♠♣❛r✱ t❡♠♦s q✉❡ b ❡ csã♦ ✐♥t❡✐r♦s✳

❙❡❣✉❡ q✉❡c2 ₌

n2+1 2

2

=

n4+2₄n2+1

.

❆❧é♠ ❞✐ss♦✱a2 ₌_n2_✱_b2 ₌

n2−1 2

2

=

₍

n

2

)

2

+ 1

2n₂2

+

n4 16

−

n2

2

+ 1 =

n4 16

+

2n

2₋_n2

2

+ 1 =

n4 16

+

n2 2

+ 1

✳

P♦rt❛♥t♦a2₊_b2 ₌_c2_✳

❊①❡♠♣❧♦ ✸✳ ❚♦♠❛♥❞♦ a=n= 7✱

t❡♠♦s q✉❡b =72_{2 = 24}−1 ❡c=72_{2 = 25}+1 ✱ s❛t✐s❢❛③❡♥❞♦ a2₊_b2 ₌_c2_✳

❊①❡♠♣❧♦ ✹✳ ❚♦♠❛♥❞♦ a=n= 6✱

t❡♠♦s q✉❡b =

(

6₂

)

2−1 = 8 ❡ c=

(

6₂

)

2+1 = 10✱ s❛t✐s❢❛③❡♥❞♦ a2+b2 =c2✳

❉❡✜♥✐çã♦ ✸✳ ❯♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦(a, b, c)é ❞❡♥♦♠✐♥❛❞♦ ♣r✐♠✐t✐✈♦ q✉❛♥❞♦ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ✐st♦ é✱mdc(a, b) = 1✳

(20)

❖❜s❡r✈❛çã♦ ✸✳ ✭✐✮ ❙❡(a, b, c)é ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦ ♣r✐♠✐t✐✈♦ ❡♥tã♦mdc(a, c) =mdc(b, c) = 1✳

✭✐✐✮ ❙❡ (a, b, c) é ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦ ❡ k é ✉♠ ✐♥t❡✐r♦✱ ❡♥tã♦ (ka, kb, kc) t❛♠❜é♠

é ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳

✭✐✐✐✮ ❙❡ (a, b, c) é ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦ ♦♥❞❡ a = ka1✱ b =kb1 ❡ c = kc1✱ k ✐♥t❡✐r♦

♥ã♦ ♥✉❧♦✱ ❡♥tã♦(a1, b1, c1)t❛♠❜é♠ é ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳

✭✐✮ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ ❡①✐st❛ ✉♠ ♣r✐♠♦p q✉❡ ❞✐✈✐❞❛a ❡ c✱ s❡❣✉❡ q✉❡p ❞✐✈✐❞❡ b2 ₌_c2₋_a2 _{❡ ♣♦rt❛♥t♦ ❞✐✈✐❞❡} _{b✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦ ♣♦✐s}_mdc₍_{a, b}_{) = 1}_✳

▲♦❣♦ mdc(a, c) = 1✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♣r♦✈❛♠♦s q✉❡ mdc(b, c) = 1✳

✭✐✐✮ ❉❡ ❢❛t♦✱(ka)2_{+ (}_kb₎2 ₌_k2₍_a2₊_b2_{) =} _k2_c2 _{= (}_kc₎2_✳

✭✐✐✐✮ ❉❡ ❢❛t♦✱ ❝♦♠♦ (ka1)2 + (kb1)2 = (kc1)2 t❡♠♦s q✉❡ k2a21 +k2b21 = k2c21 ⇒

k2₍_a2

1 +b21) =k2c21 ⇒a21+b21 =c21✳

❖❜s❡r✈❛çã♦ ✹✳ ❙❡❥❛ (a, b, c) ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳ ❈♦♥s✐❞❡r❛♥❞♦ d = mdc(a, b)✱ s❡❣✉❡ q✉❡

a=da1 ❡b =db1✱ ♦♥❞❡ (a1, b1) = 1✳

❈♦♠♦ (da1)2 + (db1)2 = c2✱ t❡♠♦s q✉❡ d2 ❞✐✈✐❞❡ c2✱ ❛ss✐♠ c2 = kd2✳ ❙❡❣✉❡

q✉❡ ✭❛♥❛❧✐s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ❢❛t♦r❡s ♣r✐♠♦s ❞♦s ✐♥t❡✐r♦s k✱ c ❡ d✮✱ k é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✱ ❞✐❣❛♠♦sk = (c1)2✱ ❛ss✐♠ t❡♠♦sc2 = (c1d)2 ❡ ❞❛í c=c1d✳ ❈♦♥❝❧✉í♠♦s q✉❡ ✉♠ t❡r♥♦

q✉❛❧q✉❡r(a, b, c)♣♦❞❡ s❡r ♦❜t✐❞♦ ❞♦ t❡r♥♦ ♣r✐♠✐t✐✈♦ (a1, b1, c1)✳ ❆ss✐♠✱ ❝♦♥❤❡❝❡♥❞♦ ♦s t❡r♥♦s

♣✐t❛❣ór✐❝♦s ♣r✐♠✐t✐✈♦s✱ ❝♦♥❤❡❝❡♠♦s t♦❞♦s ♦s ♦✉tr♦s✳

▼ét♦❞♦ ❞❡ ❊✉❝❧✐❞❡s ♣❛r❛ ❡♥❝♦♥tr❛r t❡r♥♦s ♣✐t❛❣ór✐❝♦s ♣r✐♠✐t✐✈♦s

Pr♦♣♦s✐çã♦ ✹✳ ❯♠ ♣♦♥t♦ P = (x, y) ♣❡rt❡♥❝❡♥t❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❝♦♠

r❛✐♦ ✐❣✉❛❧ ❛1t❡♠ ❝♦♦r❞❡♥❛❞❛s r❛❝✐♦♥❛✐s✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱P = (−1,0)♦✉P =

1₋t2

1+t2

,

2t

1+t2

❝♦♠ t ǫ _Q✳

❉❡♠♦♥str❛çã♦✳ (_⇐) i)❚❡♠♦s q✉❡ P = (₋1,0) ♣❡rt❡♥❝❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠

❞❡ r❛✐♦ ✐❣✉❛❧ ❛ 1✱ ♣♦✐s (0₋(₋1))2+ (0₋0)2 = 1✳

ii)❙❡t ǫQ✱ t❡♠♦s q✉❡P =

1−t2

1+t2

,

2t

1+t2

t❡♠ ❛♠❜❛s ❛s ❝♦♦r❞❡♥❛❞❛s r❛❝✐♦♥❛✐s✳

(21)

❆❧é♠ ❞✐ss♦✱ s❡❣✉❡ q✉❡

0 ₋

1₁₊−t_t22

2

+

0 ₋

)

2

=

(

1+t2

)

2

(

1+t2

)

2= 1✳ ▲♦❣♦P ♣❡rt❡♥❝❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❞❡ r❛✐♦ ✐❣✉❛❧ ❛1✳

(⇒) ❈♦♥s✐❞❡r❡♠♦s ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C ❝❡♥tr❛❞❛ ❡♠ (0,0)❞❡ r❛✐♦ 1✱ ♦ ♣♦♥t♦ P = (−1,0) ❡ ❛s r❡t❛s y = t(x+ 1)✱ ❝♦♠ t ǫ R✳ ❆s r❡t❛s ❝✐t❛❞❛s ♣❛ss❛♠ ♣♦r P = (−1,0)✱ t❡♠

✐♥❝❧✐♥❛çã♦t ❡ ❛s s✉❛s ✐♥t❡rs❡çõ❡s ❝♦♠C sã♦ ❞❛❞❛s ♣❡❧♦ s✐st❡♠❛✿

      

y=t(x+ 1) (1)

x2₊_y2 _{= 1} ₍₂₎

✱ s✉❜st✐t✉✐♥❞♦(1) ❡♠ (2) t❡♠♦s✿

x2_{+ (}_t₍_x_{+ 1))}2 _{= 1} _⇐⇒

x2₊_t2₍_x2_{+ 2}_x_{+ 1) = 1}_⇐⇒

x2₊_t2_x2_{+ 2}_t2_x₊_t2₋_{1 = 0}_⇐⇒

x2_{(1 +}_t2_{) + 2}_t2_x_{+ (}_t2₋_{1) = 0}

❙❡❣✉❡ q✉❡✿

xt=

−2t2_±q4t4₋4

(

t2+1

)(

t2₋1

)

2

(

1+t2

)

=

−2t2_±

√

4t4₋4(t4₋1) 2











1₋t2

1+t2

−

1

(3)

❙✉❜st✐t✉✐♥❞♦(3) ❡♠ (1)✱ t❡♠♦s











y

=

t

1_1+t−t22

+ 1

y

=

t

(

₋

1 + 1)

⇒











y

=

t

1₋t2₊

₍

_1+t2

₎

1+t2

y

= 0

⇒











y

=

t

_1+t2 2

y

= 0

⇒











y

=

❚♦♠❛♥❞♦ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r (₋1,0) ❡ t❡♠ ✐♥❝❧✐♥❛çã♦ t = yt

xt+1 ǫ Q✱ t❡♠♦s q✉❡ ❛ s✉❛

✐♥t❡rs❡çã♦ ❝♦♠C é ♦ ♣♦♥t♦

1₋_xtyt₊₁2

1+_xtyt₊₁2

,

2_xtyt₊₁

1+_xtyt₊₁2

!

=





(1+_xt)2−y2_t (_xt+1)2 (1+_xt)2+y2_t

(_xt+1)2

,

2_yt xt+1

(_xt+1)2+y2_t

(_xt+1)2





=

(1+xt)2−yt2

(1+xt)2+yt2

,

2yt(xt+1)2

(

(x_t+1)2+y2_t

)

(xt+1)

=

1+2xt+x2t−yt2

1+2xt+x2_t+y2_t

,

2yt(xt+1)

1+2xt+x2_t+y_t2

✱

❝♦♠♦y2

t = 1−x2t✱ t❡♠♦s q✉❡

(23)

₁₊₂_x

t+x2_t−y2_t

1+2xt+x2_t+y_t2

,

2yt(xt+1)

1+2xt+x2_t+y2_t

=

1+2xt+x2_t−

)

=

₂_x

t+2x2_t

2+2xt

,

yt2(xt+1)

2xt+2

=

xt(2+2xt)

2+2xt

,

yt(2xt+2)

2xt+2

=

(xy, yt)✳

P♦rt❛♥t♦✱ t♦❞♦ ♣♦♥t♦P ₆= (₋1,0)❝♦♠ ❛♠❜❛s ❛s ❝♦♦r❞❡♥❛❞❛s r❛❝✐♦♥❛✐s ❞❡ C é ❞❛ ❢♦r♠❛

1−t2

1+t2

,

2t

1+t2

✳

❖❜s❡r✈❛çã♦ ✺✳ ❙❡❥❛♠a, b, c ǫN ❝♦♠c6= 0✱ t❡♠♦s q✉❡a2₊_b2 ₌_c2 _⇐⇒ a c

2

+ b c

2

= 1⇐⇒

a c −0

2

+ b c −0

2

= 1✳

❖✉ s❡❥❛✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ t❡r♥♦s ♣✐t❛❣ór✐❝♦s ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❝❛r❛❝t❡✲ r✐③❛çã♦ ❞❡ ♣♦♥t♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C ❝❡♥tr❛❞❛ ❡♠(0,0)❞❡ r❛✐♦1✱ ❝♦♠ ❛♠❜❛s ❛s ❝♦♦r❞❡♥❛❞❛s

r❛❝✐♦♥❛✐s✳

Pr♦♣♦s✐çã♦ ✺✳ ❚♦❞♦s ♦s t❡r♥♦s ♣✐t❛❣ór✐❝♦s ♣r✐♠✐t✐✈♦s (a, b, c) sã♦ ❞❛❞♦s ♣♦r a = n2 −m2✱ b= 2mn✱ c=n2₊_m2_{✱ ♦♥❞❡} _mdc₍_{m, n}_{) = 1}_✱ _m _❡ _n _{t❡♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s ❡} _{m < n}_✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ a, b, c ǫ _N ❝♦♠ c ₆= 0 ❡ mdc(a, b) = 1✱ t❛✐s q✉❡ a2 ₊ _b2 ₌

c2_{✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ✺ ❡ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✹✱ t❡♠♦s q✉❡} a

c

,

b c

=

1₁₊−t_t22

,

₁₊2t_t2

=

1−

(

mn

)

n2 n2+m2

n2

,

2mn n2+m2

n2

=

n_n22−₊_mm22

,

_n22₊mn_m2

✱ ♦♥❞❡ ❝♦♥s✐✲ ❞❡r❛♠♦st= m

n ❝♦♠ mdc(m, n) = 1✳

❉❛ ✐❣✉❛❧❞❛❞❡ ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s✱ t❡♠♦s a_c

=

m_n22₊−_mn22 ❡ b_c

=

_n22₊mn_m2✳ ❈♦♠♦

mdc(a, b) = 1 ❡ a2₊_b2 ₌_c2_{✱ ❝♦♥❝❧✉í♠♦s q✉❡} _mdc₍_{a, c}_{) = 1} _❡ _mdc₍_{b, c}_{) = 1} _{✭♦❜s❡r✈❛çã♦ ✸✮✳}

❈♦♠♦mdc(m, n) = 1✱ t❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿

✶✮m ❡ n t❡♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✳

◆❡st❡ ❝❛s♦✱mdc(m2₋_n2_{, n}2₊_m2_{) = 1} _❡ _mdc₍₂_{mn, m}2₊_n2_{) = 1}_✳

❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡mdc(m2₋_n2_{, n}2₊_m2₎₆_{= 1}_{✳ ❈♦♥s✐❞❡r❡}

p♣r✐♠♦ q✉❡ ❞✐✈✐❞❡n2₋_m2 _❡_n2₊_m2_{✳ ❈♦♠♦}_m_❡_n_{t❡♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✱ t❡♠♦s q✉❡}_n2₋_m2_❡

n2₊_m2_{sã♦ í♠♣❛r❡s✱ ♣♦rt❛♥t♦}_p₆_{= 2}_{✳ ❆❧é♠ ❞✐ss♦✱}_p_{❞✐✈✐❞❡ ❛ s♦♠❛}₍_n2₋_m2₎₊₍_n2₊_m2_{) = 2}_n2

(24)

❡ ❛ ❞✐❢❡r❡♥ç❛(n2 ₊_m2₎₋₍_n2₋_m2_{) = 2}_m2_{✳ ▲♦❣♦✱} _p_{❞✐✈✐❞❡} _m _❡_{n✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱}

♣♦✐sm ❡ n sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ mdc(2mn, n2₊_m2₎ ₆_{= 1}_{✳ ❈♦♥s✐❞❡r❡} _p _{♣r✐♠♦ q✉❡ ❞✐✈✐❞❡}

2mn ❡n2₊_m2_{✳ ❈♦♠♦} _n2₊_m2 _{é í♠♣❛r✱ t❡♠♦s q✉❡}_p₆_{= 2}_{✳ ❆ss✐♠✱} _p₆_{= 2}_❡ _p_{❞✐✈✐❞❡} ₂_{mn✱ ❧♦❣♦}

p ❞✐✈✐❞❡ m ♦✉ p ❞✐✈✐❞❡ n✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛♠♦s q✉❡ p ❞✐✈✐❞❡ m✱ s❡❣✉❡ q✉❡ p ❞✐✈✐❞❡ m2_{✱ ❝♦♠♦} _p _{❞✐✈✐❞❡} _n2₊_m2_{✱ ❝♦♥❝❧✉í♠♦s q✉❡} _p _{❞✐✈✐❞❡} _n2 _{❡ ♣♦rt❛♥t♦ ❞✐✈✐❞❡} _n_{✱ ♦}

q✉❡ é ♥♦✈❛♠❡♥t❡ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐sm ❡ n sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥❛s ✐❣✉❛❧❞❛❞❡s a_c

=

m_n22₊−_mn22 ❡ b_c

=

_n22mn₊_m2 t♦❞❛s ❛s ❢r❛çõ❡s sã♦ ✐rr❡❞✉tí✈❡✐s✳ P♦rt❛♥t♦

a=m2₋_n2_✱ _b _{= 2}_mn _❡_c₌_n2₊_m2_✳

✷✮m ❡ n sã♦ ❛♠❜♦s í♠♣❛r❡s✿

❈♦♥s✐❞❡r❡ p=m+₂n ❡q =n−₂m✱ t❡♠♦s q✉❡ p❡ q sã♦ ✐♥t❡✐r♦s ♣r✐♠♦s ❡♥tr❡ s✐ ❝♦♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✳ ❙❡ ❡①✐st✐ss❡ ✉♠ ♥❛t✉r❛❧ ❞✐✈✐s♦r ❝♦♠✉♠ ❞✐❢❡r❡♥t❡ ❞❡ 1 q✉❡ ❞✐✈✐❞✐ss❡ p ❡ q✱ ❡st❡ ♥❛t✉r❛❧ ❞✐✈✐❞✐r✐❛ ❛ s♦♠❛ (n) ❡ ❛ ❞✐❢❡r❡♥ç❛ (m) ❡♥tr❡ ❡❧❡s ✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ❙❡

t✐✈❡ss❡♠ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡✱ 2 ❞✐✈✐❞✐r✐❛ ❛ s♦♠❛ (n) ❡ ❛ ❞✐❢❡r❡♥ç❛ (m) ❡♥tr❡ ❡❧❡s✱ ♦ q✉❡ é

♥♦✈❛♠❡♥t❡ ✉♠ ❛❜s✉r❞♦✳

❯s❛♥❞♦ p =m+₂n_⇐⇒ 2p = m+n ❡ q =n−₂m_⇐⇒ 2q = n₋m ❡♠ a_c

,

b_c

=

n2₋m2 n2+m2

,

2mn n2+m2

(2q)(2p)

(p+q)2+(p₋q)2

,

2(p−q)(p+q) (p+q)2+(p₋q)2

=

(2q)(2p) 2

(

p2₊_q2

₎

,

2(p−q)(p+q) 2

(

p2₊_q2

₎

=

_p22₊pq_q2

,

p2₋q2 p2+q2

✱ ❝♦♠ p ❡ q ❝♦♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s ❡ mdc(p, q) = 1✱ ♦ q✉❡

♥♦s ❢❛③ r❡t♦r♥❛r ❛♦ ❝❛s♦ ✶✮✳ P♦rt❛♥t♦✱ é ❧❡❣ít✐♠♦ t♦♠❛ra= 2pq✱b =p2₋_q2 _❡_c₌_p2₊_q2_✳

❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s q✉❡ ❡ss❛ ♠áq✉✐♥❛ ❞❡ t❡r♥♦s ♣✐t❛❣ór✐❝♦s ❝♦♠ ❡❧❡♠❡♥t♦s ♣r✐♠♦s ❡♥tr❡ s✐ ❞♦✐s ❛ ❞♦✐s ♥♦s ❢♦r♥❡❝❡✿

❊①❡♠♣❧♦ ✺✳ ❚♦♠❛♥❞♦ t= 1

2✱ t❡♠♦s q✉❡ a= 2

2 ₋₁2 _{= 3}_✱ _b_{= 2}_.₁_._{2 = 4} _❡ _c_{= 2}2_{+ 1}2 _{= 5}_✳

(25)

❊①❡♠♣❧♦ ✻✳ ❚♦♠❛♥❞♦ t = 3₇✱ ❞❡✈❡♠♦s t♦♠❛r p = 3+7₂ = 5 ❡ q = 7−3

2 = 2✳ ❆ss✐♠✱ t❡♠♦s

a= 2.5.2 = 20✱b = 52₋₂2 _{= 21} _❡ _c_{= 5}2_{+ 2}2 _{= 29}_✱

♦❜t❡♥❞♦ 292 _{= 20}2_{+ 21}2_{✱ ❝♦♠} _mdc₍₂₀_,_{21) = 1}_.

✸✳✷ Pr✐♠♦s ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s

Pr♦♣♦s✐çã♦ ✻✳ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ í♠♣❛r ❡ p=a2₊_b2_{✱ ❡♥tã♦} _p_{= 4}_k_{+ 1} _❝♦♠ _{k ǫ} _N_✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s três ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿ a ❡ b ♣❛r❡s✱ a ❡ b í♠♣❛r❡s ♦✉ a ❡ b ❝♦♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✱

(i) ❙❡a ❡b ❢♦ss❡♠ ❛♠❜♦s ♣❛r❡s✱ t❡rí❛♠♦s q✉❡ a2₊_b2_✭₌_p_{✮ s❡r✐❛ ✉♠ ♥ú♠❡r♦ ♣❛r✱}

❝♦♥tr❛r✐❛♥❞♦ ❛ ❤✐♣ót❡s❡✳

(ii) ❙❡ a ❡ b ❢♦ss❡♠ ❛♠❜♦s í♠♣❛r❡s✱ ♥♦✈❛♠❡♥t❡ t❡rí❛♠♦s q✉❡ a2 ₊_b2_✭₌ _p_{✮ s❡r✐❛}

✉♠ ♥ú♠❡r♦ ♣❛r✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ ❤✐♣ót❡s❡✳

(iii) ❙❡ a ❡ b t❡♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✱ s✉♣♦♥❤❛♠♦s s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ a= 2k ✉♠ ♥ú♠❡r♦ ♣❛r ❡b = 2t+ 1✉♠ ♥ú♠❡r♦ í♠♣❛r✱ t❡♠♦s q✉❡a2 _{= 4}_k2 _❡_b2 _{= 4}_t2_{+ 4}_t_{+ 1}

❡ a2 ₊_b2 _{= 4}_k2 _{+ (4}_t2_{+ 4}_t_{+ 1) = 4 (}_k2₊_t2 ₊_t_{) + 1}_{✳ P♦rt❛♥t♦} _p ₌ _a2 ₊_b2 _{é ❞❛ ❢♦r♠❛}

4k+ 1✳

P❛r❛ ❝❛❞❛ ♥❛t✉r❛❧n✱ s❡❥❛r(n)♦ ♥ú♠❡r♦ ❞❡ ♠♦❞♦s ❞✐st✐♥t♦s ❞❡ s❡ ❡s❝r❡✈❡rn ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ n =x2₊_y2_{✱ ❝♦♠} _x _❡ _y _{✐♥t❡✐r♦s✳ ❆♦ ❝❛❧❝✉❧❛r♠♦s} _r₍_n₎_{✱ ♣❡♥s❛r❡♠♦s}

♥❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s(a, b) ❞❡n =x2₊_y2 _{❝♦♠♦ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ ✐♥t❡✐r♦s✳ P♦r ❡①❡♠♣❧♦✱}

8 = 22_{+ (}₋₂₎2 _❡ _{8 = (}₋₂₎2_{+ 2}2_{✱ sã♦ ❞✉❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❞❡ ❡s❝r❡✈❡r} ₈ _❝♦♠♦

s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✿ ❊①❡♠♣❧♦ ✼✳ r(8) = 4✱ ♣♦✐s

8 = 22_{+ 2}2

8 = (−2)2+ (−2)2

(26)

8 = (₋2)2_{+ 2}2

8 = 22_{+ (}₋₂₎2

❊①❡♠♣❧♦ ✽✳ r(10) = 8✱ ♣♦✐s

10 = 32_{+ 1}2 _{= 1}2_{+ 3}2 _{= (}₋₁₎2 _{+ 3}2 _{= 3}2_{+ (}₋₁₎2 _{= 1}2 _{+ (}₋₃₎2 _{= (}₋₃₎2_{+ 1}2 ₌

(₋1)2_{+ (}₋₃₎2 _{= (}₋₃₎2_{+ (}₋₁₎2

❊①❡♠♣❧♦ ✾✳ r(17) = 8✱ ♣♦✐s

17 = 12+ 42 = 42+ 12 = (−1)2 + 42 = 42+ (−1)2 = 12 + (−4)2 = (−4)2+ 12 = (₋1)2_{+ (}₋₄₎2 _{= (}₋₄₎2_{+ (}₋₁₎2_✳

❖❜s❡r✈❛♠♦s q✉❡ ♦ ♣r✐♠♦ 2 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ♣♦✐s 2 = 12_{+ 1}2_{✳ ❆❧é♠ ❞✐ss♦✱}_r_{(2) = 4}_{✱ ❥á q✉❡ ❛s ú♥✐❝❛s ❡s❝r✐t❛s ❞❡}₂_{❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s}

sã♦ 2 = (₋1)2+ (₋1)2✱ 2 = (₋1)2 + 12_✱ _{2 = 1}2 _{+ (}₋₁₎2 _❡

2 = 12 _{+ 1}2_{✳ ❖ ♣r✐♠♦} ₅ _{t❛♠❜é♠}

♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ♣♦✐s5 = 12_{+ 2}2 _{= (}₋₁₎2

+ 22 _{= 1}2_{+ (}₋₂₎2

=

(−1)2+ (−2)2 = 22_{+ 1}2 _{= (}₋₂₎2_{+ 1}2 _{= 2}2_{+ (}₋₁₎2 _{= (}₋₂₎2_{+ (}₋₁₎2_{✱ ♣♦rt❛♥t♦} _r_{(5) = 8}_{✳ ❏á}

♦ ♣r✐♠♦3 ♥ã♦ ♣♦❞❡ t❡r t❛❧ ❡s❝r✐t❛✱ ❧♦❣♦r(3) = 0✳

❖❜s❡r✈❛çã♦ ✻✳ ❖❜s❡r✈❛♠♦s q✉❡ s❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ í♠♣❛r ❡ p=a2₊_b2_{✱ ❡♥tã♦} _a ₆₌_b _❡

ab₆= 0✳

❉❡ ❢❛t♦✱ s❡a=b✱ t❡rí❛♠♦s q✉❡p= 2a2_{✱ ♦✉ s❡❥❛✱ t❡rí❛♠♦s q✉❡}_p_{é ✉♠ ♥ú♠❡r♦ ♣❛r✳}

❙❡a= 0 ♦✉ b= 0✱ t❡rí❛♠♦s p=a2 _♦✉ _p₌_b2_{✱ q✉❡ ♥ã♦ sã♦ ♣r✐♠♦s✳}

▲❡♠❛ ✹✳ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ í♠♣❛r ❡ p=a2+b2✱ ❡♥tã♦ r(p) = 8✳

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ♦❜s❡r✈❛çã♦ ✻ ♥ós ❝♦♥❝❧✉í♠♦s q✉❡a₆=b✱ ❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡rp❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ❞❡ ♣❡❧♦ ♠❡♥♦s ✽ ♠❛♥❡✐r❛s✱ ✉s❛♥❞♦ ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞♦ ❝♦♥❥✉♥t♦ X =_{(a, b),(₋a, b),(a,₋b),(₋a,₋b),(b, a),(₋b, a),(b,₋a),(₋b,₋a)_}✳

❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛(c, d)∈/ X✱ t❛❧ q✉❡ p=a2₊_b2 ₌_c2₊_d2_{✳ ❈♦♠♦} _p_{é í♠♣❛r✱}

a❡btê♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s ❡c❡dt❛♠❜é♠ tê♠ ♣❛r✐❞❛❞❡s ♦♣♦st❛s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ s✉♣♦♥❤❛♠♦sa❡c♣❛r❡s✱ ❧♦❣♦b❡dí♠♣❛r❡s✳ ❚❡♠♦s q✉❡a2+b2 =c2+d2✱ ❧♦❣♦a2−c2 =d2−b2✱

(27)

❛ss✐♠ (a₋c) (a+c) = (d₋b) (d+b) ✭✶✮✳ ❈♦♠♦ ❛ s♦♠❛ ♦✉ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♥ú♠❡r♦s ❞❡

♠❡s♠❛ ♣❛r✐❞❛❞❡ r❡s✉❧t❛ ❡♠ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ❝♦♥❝❧✉í♠♦s q✉❡(a₋c)✱(a+c)✱(d₋b)❡(d+b)

sã♦ t♦❞♦s ♥ú♠❡r♦s ♣❛r❡s✳ ❈♦♠♦ c 6= ±a ❡ d 6= ±b✱ ❝♦♥s✐❞❡r❛♥❞♦ D = mdc(a−c, d−b) ❡

E =mdc(a+c, d+b)✱ s❡❣✉❡ q✉❡ D ❡E sã♦ ❛♠❜♦s ♥ú♠❡r♦s ♣❛r❡s✱ ❡ ❡①✐st❡♠✿ i)l1✱l2 ǫ N✱ t❛✐s q✉❡a−c=l1D ❡d−b =l2D ✭✷✮✱ ♦♥❞❡ mdc(l1, l2) = 1✳

ii)k1✱k2 ǫ N✱ t❛✐s q✉❡ a+c=k1E ❡ d+b=k2E ✭✸✮✱ ♦♥❞❡ mdc(k1, k2) = 1✳

❉❡ ✭✶✮✱ ✭✷✮ ❡ ✭✸✮ t❡♠♦s✱ (a₋c) (a+c) = (d₋b) (d+b) _⇒l1Dk1E =l2Dk2E ⇒

l1k1 = l2k2 ⇒ _kk1₂ = l_l2₁✳ ◆❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ t❡♠♦s ❞✉❛s ❢r❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s ♥❛s s✉❛s

❢♦r♠❛s ✐rr❡❞✉tí✈❡✐s✱ ♣♦rt❛♥t♦✱ t❡♠♦sk1 =l2 ❡ k2 =l1 ✭✹✮✳

❆ss✐♠✱ t❡♠♦s q✉❡✱

a−c=l1D ❡ d−b=l2D ✭✷✮✳

a+c=l2E ❡ d+b =l1E ✭✸✮ ❡ ✭✹✮✳

❙❡❣✉❡ q✉❡✱ (a₋c) + (a+c) = l1D+l2E ⇒2a=l1D+l2E ⇒a=

l₁D+l₂E

2 ❡✱

(d+b)₋(d₋b) =l1E−l2D⇒2b=l1E−l2D⇒b =

l₁E₋l₂D

2

✳

❉❛í✱ p=a2₊_b2 ₌

l₁D+l₂E

2

+

l1E−l2D

2

=

l21D2+2l1Dl2E+l22E2

4

+

l2₁E2−2l1Dl2E+l2₂D2

4

=

l2₁D2+l₁2E2+l₂2E2+l2₂D2

4

=

l2₁

(

D2+E2

)

+l2₂

(

D2+E2

)

4

=

(

l2₁+l₂2

)(

D2+E2

)

4

=

(l2 1+l22)

(

D2+E2

)

4

=

l

2₁

+

l

₂2

h

_D2

4

+

E2

4

i

=

l

2₁

+

l

2₂

h

_D

2

+

E 2

2

i

✳ ❈♦♠♦ (l2

1+l22) ❡

h D 2 2 + E 2 2i

sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s ♠❛✐♦r❡s q✉❡1✱ t❡rí❛♠♦s p ❝♦♠♣♦st♦✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦✱ s❡p♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✱ ❡♥tã♦ r(p) = 8✳

❉❡♠♦♥str❛♠♦s q✉❡ s❡p=a2₊_b2 _❝♦♠_p_{♣r✐♠♦✱ ❡♥tã♦}_p_{= 4}_k_{+ 1}_❝♦♠_{k ǫ}_N_{✭❞❡✐①❛}

r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 4✮✱ ❧♦❣♦ ♣r✐♠♦s ❞❛ ❢♦r♠❛ 4k + 3 ♥ã♦ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦

s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ♣r♦✈❛♠♦s q✉❡ s❡p=a2₊_b2 _❡♥tã♦ _r₍_p_{) = 8}_✳

(28)

❖ t❡♦r❡♠❛ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❣r❛♥❞❡ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ❛✜r♠❛ q✉❡ t♦❞♦ ♣r✐♠♦ p ❞❛ ❢♦r♠❛4k+ 1 ♣♦❞❡ ❞❡ ❢❛t♦ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ❡ ♣♦rt❛♥t♦r(p) = 8✳

P❛r❛ ❝♦♠♣❧❡t❛r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ✉s❛r❡♠♦s ✉♠ t✐♣♦ ❞❡ ❢✉♥çã♦✱ ❝❤❛♠❛❞❛ ✐♥✈♦❧✉çã♦✱ ❞❡✜♥✐❞❛ ❛ s❡❣✉✐r✳

❉❡✜♥✐çã♦ ✹✳ ❙❡❥❛S✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ ✉♠❛ ❢✉♥çã♦f :S→Sé ✉♠❛ ✐♥✈♦❧✉çã♦ s❡f of =IS✱

♦♥❞❡IS :S →S é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✳

❖❜s❡r✈❛♠♦s q✉❡ ❛ ❝♦♥❞✐çã♦ f of = IS é ❡q✉✐✈❛❧❡♥t❡ ❛ ❛✜r♠❛çã♦ ✏f é ❜✐❥❡t✐✈❛ ❡

❝♦✐♥❝✐❞❡ ❝♦♠ s✉❛ ✐♥✈❡rs❛✑✳

❉❡✜♥✐çã♦ ✺✳ ❯♠ ♣♦♥t♦ ✜①♦ ❞❡ ✉♠❛ ❢✉♥çã♦f : S _→ S✱ é ✉♠ ♣♦♥t♦x0 t❛❧ q✉❡ f(x0) =x0✳

Pr♦♣♦s✐çã♦ ✼✳ ❙❡❥❛ S ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ f ✉♠❛ ✐♥✈♦❧✉çã♦✳ ❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ S ❡ ♦ ♥ú♠❡r♦ ❞♦s ♣♦♥t♦s ✜①♦s ❞❡ f tê♠ ♠❡s♠❛ ♣❛r✐❞❛❞❡✳

❉❡♠♦♥str❛çã♦✳ Pr♦✈❛r❡♠♦s ❡ss❛ ♣r♦♣♦s✐çã♦ ♣♦r ✐♥❞✉çã♦✳ ❙✉♣♦♥❤❛ q✉❡S t❡♥❤❛n ❡❧❡♠❡♥t♦s ❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✜①♦s ❞❡ f ❡♠ S s❡❥❛ ❞❡s✐❣♥❛❞♦ ♣♦rFfS✳

P❛ss♦ ✶✮ ❙❡n = 1✱ t❡♠♦s q✉❡ S =_{a1} ❡f(a1) = a1✳ ❆ss✐♠ n= 1 =FfS✳

❙❡ n = 2 ✭S = _{a1, a2}✮✱ ❝♦♠♦ f é ✉♠❛ ✐♥✈♦❧✉çã♦ t❡♠♦s ❛♣❡♥❛s ❞✉❛s ♣♦ss✐❜✐❧✐✲

❞❛❞❡s✿ f(a1) = a1 ❡ f(a2) = a2 ♦✉ f(a1) = a2 ❡ f(a2) = a1✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦ FfS t❡♠ ✷

❡❧❡♠❡♥t♦s✱ ♥♦ s❡❣✉♥❞♦ ❝❛s♦FfS t❡♠ ✵ ❡❧❡♠❡♥t♦s✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦sFfS t❡♠ ♠❡s♠❛ ♣❛r✐❞❛❞❡

q✉❡S✳

P❛ss♦ ✷✮ ❙✉♣♦♥❤❛♠♦s ❛ ♣r♦♣♦s✐çã♦ ✈á❧✐❞❛ ♣❛r❛ q✉❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ t❡♥❤❛ ❛tén ❡❧❡♠❡♥t♦s✱ t❡♠♦s q✉❡ ♠♦str❛r q✉❡ ♦ ♠❡s♠♦ é ✈á❧✐❞♦ ♣❛r❛ q✉❛♥❞♦ f t❡♥❤❛ n+ 1 ❡❧❡♠❡♥t♦s✳

❙❡❥❛♠ S =_{a1, a2, ..., an+1} ❡f : S → S ✉♠❛ ✐♥✈♦❧✉çã♦✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s✿

i) f(an+1) = an+1✳ ❈♦♥s✐❞❡r❡ f r❡str✐t❛ ❛♦ ❝♦♥❥✉♥t♦ S1 =S− {an+1}✱ ❝♦♠ ❡st❛

r❡str✐çã♦ f ❝♦♥t✐♥✉❛ s❡♥❞♦ ✉♠❛ ✐♥✈♦❧✉çã♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ S1 ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✜①♦s ❞❡ f ❡♠ S1 t❡♠ ✐❣✉❛❧ ♣❛r✐❞❛❞❡✳ ❙❡ S1 ❡ FfS₁ t❡♠

q✉❛♥t✐❞❛❞❡ í♠♣❛r ❞❡ ❡❧❡♠❡♥t♦s✱ ❡♥tã♦S ❡ FfS t❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s✳ ❙❡ S1

❡FfS₁ t❡♠ q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s✱ ❡♥tã♦S ❡FfS t❡♠ q✉❛♥t✐❞❛❞❡ í♠♣❛r ❞❡ ❡❧❡♠❡♥t♦s✳

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ii)f(an+1) =ak✱ ❝♦♠1≤k ≤n✳ ❈♦♠♦f :S→Sé ✉♠❛ ✐♥✈♦❧✉çã♦✱f(ak) =an+1✳

❈♦♥s✐❞❡r❡ S2 = S − {ak, an+1}✱ ❝♦♠ ❡st❛ r❡str✐çã♦ f ❝♦♥t✐♥✉❛ s❡♥❞♦ ✉♠❛ ✐♥✈♦❧✉çã♦✱ ♣❡❧❛

❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ t❡♠♦s q✉❡ S2 ❡ FfS₂ t❡♠ ✐❣✉❛❧ ♣❛r✐❞❛❞❡✳ ❚❡♠♦s q✉❡ S2 t❡♠ ♠❡s♠❛

♣❛r✐❞❛❞❡ q✉❡S✱ ❝♦♠♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣♦♥t♦s ✜①♦s ♥ã♦ ♠✉❞❛rá ♥❛ ♣❛ss❛❣❡♠ ❞♦ ❞♦♠í♥✐♦ ❞❡ S2 ♣❛r❛ S, ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ S ❡FfS t❡♠ ✐❣✉❛❧ ♣❛r✐❞❛❞❡✳

❚❡♦r❡♠❛ ✺✳ ✭●r❛♥❞❡ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮ ❙❡ p é ♣r✐♠♦ ❞❛ ❢♦r♠❛ 4k+ 1 ❝♦♠ k ǫ _N✱ ❡♥tã♦ r(p) = 8✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❞❛ ❢♦r♠❛ 4k + 1✱ ❝♦♠ k ǫ N✳ ❈♦♥s✐❞❡r❡♠♦s ♦

❝♦♥❥✉♥t♦ S = {(x, y, z)ǫN3_\_x2_{+ 4}_yz ₌_p_}✳ _S _{é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ ♣♦✐s} ₍₁_,₁_{, k}₎ _{ǫ S✳}

❆❧é♠ ❞✐ss♦✱ S é ✜♥✐t♦✱ ❥á q✉❡ ♦s ✈❛❧♦r❡s ❞❡ x, y ❡ z ❡stã♦ ❡♥tr❡ 1 ❡ p✳

❙❡❥❛f :S → S ✉♠❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦r

f(x, y, z) =

                

(x+ 2z, z, y−x−z)_{, se} x < y−z

(2y−x, y, x−y+z)_{, se} y−z < x <2y

(x₋2y, x₋y+z, y)_{, se} x >2y

❆ ❢✉♥çã♦f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ❥á q✉❡ ♦s ♣❧❛♥♦sx=y₋z❡x= 2y♥ã♦ ✐♥t❡rs❡❝t❛♠ S✭s❡ ✐♥t❡rs❡❝t❛ss❡♠✱ t❡rí❛♠♦s ❡❧❡♠❡♥t♦s ❞❡Ss❡♠ ❝♦rr❡s♣♦♥❞❡♥t❡✮✳ ❉❡ ❢❛t♦✱ s✉❜st✐t✉✐♥❞♦x=

y−z❡♠x2₊₄_yz ₌_{p✱ t❡rí❛♠♦s}₍_y₋_z₎2₊₄_yz ₌_p_⇒_y2₋₂_yz₊_z2₊₄_yz ₌_p_⇒_y2₊₂_yz₊_z2 ₌

p⇒(y+z)2 = p✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s p é ♣r✐♠♦ ❡ ♥ã♦ ♣♦❞❡ s❡r q✉❛❞r❛❞♦ ❞❡ ♥❡♥❤✉♠ ♥❛t✉r❛❧✳ ❙✉❜st✐t✉✐♥❞♦ x = 2y ❡♠ x2_{+ 4}_yz ₌ _p_{✱ t❡rí❛♠♦s} _p _{= (2}_y₎2

+ 4yz = 4 (y2₊_yz₎_{✱ ♦}

q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s p é ♣r✐♠♦ ❞❛ ❢♦r♠❛ 4k+ 1✳ ❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡

f r❡❛❧♠❡♥t❡ ❡stá ❡♠ S✱ ♣♦✐s (x+ 2z)2+ 4 (z) (y₋x₋z) = (2y₋x)2 + 4 (y) (x₋y+z) = (x₋2y)2+ 4 (x₋y+z) (y) =x2_{+ 4}_yz✳

❱❛♠♦s ♣r♦✈❛r ❛❣♦r❛ q✉❡ ❛ ❢✉♥çã♦f é ✉♠❛ ✐♥✈♦❧✉çã♦✳ ❚❡♠♦s q✉❡

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f(x, y, z) =                   

(x+ 2z, z, y−x−z) , _se (x, y, z) ǫ S1

(2y₋x, y, x₋y+z) , _se (x, y, z) ǫ S2

(x₋2y, x₋y+z, y) , _se (x, y, z) ǫ S3

♦♥❞❡ S1 = {(x, y, z)ǫS/x < y−z}✱ S2 = {(x, y, z)ǫS/y−z < x <2y} ❡ S3 =

{(x, y, z)ǫS/x >2y_}✳

❖❜s❡r✈❛♠♦s q✉❡f(S1)⊂S3✱ f(S3)⊂S1 ❡ f(S2)⊂S2✱ ❧♦❣♦

f(f(x, y, z)) =

              

((x+ 2z)₋2 (z),(x+ 2z)₋(z) + (y₋x₋z), z) (2y₋(2y₋x), y,(2y₋x)₋(y) + (x₋y+z)) (x₋2y) + 2 (y), y,(x₋y+z)₋(x₋2y)₋(y)

=               

(x, y, z) (x, y, z) (x, y, z)

❆✜r♠❛çã♦✿ 1,1,p−₄1

ǫ S2 é ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❞❡ f✳

Pr✐♠❡✐r❛♠❡♥t❡✱f 1,1,p−₄1

= 2−1,1,1−1 + p−₄1

= 1,1,p−₄1

✳ ❯♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ✜①♦✿

❈♦♠♦ f(S1) ⊂ S3 ❡ f(S3) ⊂ S1✱ ❛ ❢✉♥çã♦ f ♥ã♦ ♣♦ss✉✐ ♣♦♥t♦s ✜①♦s ❡♠ S1 ❡ S3

✭S1∩S3=∅✮✳

❙✉♣♦♥❤❛♠♦s ❡♥tã♦(x, y, z)ǫ S2 t❛❧ q✉❡ f(x, y, z) = (x, y, z)✱ ♦✉ s❡❥❛✱

(2y−x, y, x−y+z) = (x, y, z)✳ ❆ss✐♠✱

              

2y₋x=x y=y

x₋y+z =z

❡ ❞❛í ♦❜t❡♠♦s x=y✳

❆ss✐♠✱ ♦ ♣♦ssí✈❡❧ ♣♦♥t♦ ✜①♦ é ❞❛ ❢♦r♠❛ (x, x, z) ǫ S2 ⊂S✳ ❙❡❣✉❡ q✉❡ x2+ 4xz =

p⇒p=x(x+ 4z)✳ ❈♦♠♦p é ♣r✐♠♦✱ ❝♦♥❝❧✉í♠♦s q✉❡x= 1✱ ❧♦❣♦p= 4z+ 1✱ ✐st♦ é✱z = p−₄1✳

P♦rt❛♥t♦✱ (x, y, z) = 1,1,p−1 4

é ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦✳