コラッツ予想について

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1. ίϥοπࣸ૾ʹ͓͚Δෆنଇيಓͷ୅਺తҰൠࣜɺ LTEิ୊ʹجͮ͘ 2ਐ෇஋ධՁɺ͓ΑͼऩଋΛ્Ήೋେܾఆ࿦ త՝୊ͷಉఆ ౦෦தֶߍ 2 ೥ ย෵ ᖕ 2026 ೥ 5 ݄ 14 ೔ ֓ཁ ຊߘͰ͸ɺίϥοπ༧૝ʹ͓͚Δ೚ҙͷෆنଇͳيಓΛ׬શʹࢧ഑͢Δ୅਺తҰൠࣜΛఆࣜԽ͠ɺ਺͕طଘͷऩ ଋϋϒ΁ͱ߹ྲྀ͢Δܾఆ࿦తͳ਺ཧߏ଄Λղ๤͢Δɻ·ͣɺྦྷੵ͞Εͨඍௐ੔߲ S ͕ S ̸≡ 0 (mod 3) Ͱ͋Δ୅ ਺తෆมੑΛূ໌͢Δɻ࣍ʹɺॳظ஋ 27 ͓Αͼ 97 ͷ࣮σʔλ͔Βɺෆنଇيಓ͕طଘͷ෼ذϋϒ໢΁ڧ੍߹ྲྀ͞ ͤΒΕΔػߏΛ୅਺తʹղ๤͢Δɻ͞Βʹɺيಓͷ௖఺Ͱൃಈ͢Δ 2 ਐ෇஋ݶքϒϨʔΩͷڍಈΛɺ਺࿦ʹ͓͚Δ LTE ิ୊ʢLifting the Exponent LemmaʣΛ༻͍ͯݫີʹݕূ͢Δɻ࠷ޙʹɺ׬શղܾΛ્Ήೋେܾఆ࿦త೉ นͱͯ͠ɺແݶൃࢄΛ൱ఆ͢ΔͨΊͷʮح਺εςοϓʹ͓͚ΔϒϨʔΩܹݮͷ՝୊ʯ ɺ͓ΑͼผϧʔϓΛ൱ఆ͢Δ ͨΊͷʮ෼ࢠ෼฼ͷ p ਐτϙϩδʔతඇಉظͷ՝୊ʯͷ 2 ఺Λਖ਼֬ʹಛఆ͠ɺղܾ΁ͷϩʔυϚοϓΛఏࣔ͢Δɻ 1
2. 1 ίϥοπيಓͷ୅਺తҰൠࣜͷఆࣜԽ ਖ਼ͷح਺ x ʹର͢ΔγϣʔτΧοτ൛ίϥοπ ࣸ૾ T(x) = (3x + 1)/2ɺ͓Αͼۮ਺ʹର͢Δࣸ૾ T(x) = x/2 Λఆٛ͢Δ [Lagarias1985]ɻ೚ҙͷॳظ ஋ x0 ∈ N ͔Βग़ൃ͠ɺ్தͰح਺ૢ࡞Λ߹ܭ A ճɺ ۮ਺ૢ࡞Λ߹ܭ B ճߦͬͨஈ֊ͷ౸ୡ਺ xN ʢͨͩ ͠ N = A + Bʣ͸ɺҎԼͷ୯ҰͷํఔࣜʹΑͬͯ׬ શʹࢧ഑͞ΕΔɻ xN = 3A · x0 + S 2A+B (1) ͜͜ͰɺS ͸֤ح਺ૢ࡞ͷλΠϛϯάʹ͓͍ͯൃੜ͢ Δ༨ΓͷྦྷੵͰ͋Γɺ֤ح਺ૢ࡞ of ޙͷۮ਺ૢ࡞ͷ ྦྷੵճ਺Λ m1 , m2 , . . . , mA ͱ͢ΔͱɺҎԼͷҰҙͳ ୅਺࿨ͱͯ͠ҰൠԽ͞ΕΔɻ S = A ∑ i=1 3A−i · 2mi (2) 2 ඍௐ੔߲ S ͷϞδϡϥʔෆมੑূ໌ ਺͕طଘͷϋϒʹিಥ͢Δ࢓૊ΈΛղ໌͢ΔͨΊɺ ඍௐ੔߲ S ͕࣋ͭ୅਺తੑ࣭Λݕূ͢Δɻ ิ୊ 1. ೚ҙͷح਺ɾۮ਺ͷग़ݱཤྺʹ͓͍ͯɺྦྷੵ ͞Εͨඍௐ੔߲ S ͸ઈରʹ 3 ͷഒ਺ʹͳΓಘͳ͍ɻ S ̸≡ 0 (mod 3) (3) Proof. ࣜ (2) ͷల։߲Λɺ3 Λ๏ͱ͢Δ߹ಉࣜʢ (mod 3)ʣͰධՁ͢Δɻ S = 3A−1·2m1 +· · ·+30·2mA ≡ 2mA (mod 3) (4) 2 ͷྦྷ৐਺Λ 3 Ͱׂͬͨ༨Γ͸ɺࢦ਺ͷۮحʹԠ ͯ͡ඞͣ 1 ·ͨ͸ 2 ͷ͍ͣΕ͔ͱͳΔͨΊɺS ≡ 1 ·ͨ͸ 2 (mod 3) ͱͳΓɺ༨Γ͕ 0 ʹͳΔ͜ͱ͸ ઈରʹͳ͍ɻ 3 ݶքϒϨʔΩɺ৒༨ྨτϥοϓɺ͓Αͼ ೖग़ྗύϦςΟରশੑ 3.1 2 ਐ෇஋ v2 ʹΑΔݶքϒϨʔΩͷҰൠԽ ੔਺ n ΛׂΓ੾Δ࠷ߴͷ 2 ͷྦྷ৐਺Λද͢ 2 ਐ ෇஋ v2 (n) Λಋೖ͢Δɻܭࢉ݁Ռͱͯ͠ੜ੒͞Εͨ ੔਺ xN ͕ v2 (xN ) > 1 Λຬͨͨ͠ॠؒɺֻ͚ࢉ ʹΑΔ૿େ͸ϩοΫ͞ΕɺҰํతͳۮ਺ফඅϑΣʔ ζʢϒϨʔΩϋϒʣ΁ͱڧ੍Ҡߦ͢Δɻॳظ஋ 27 ʹ ͓͚Δ 45 ख໨ͷ࠷ߴ஋ 4616 ΛૉҼ਺෼ղ͢Δͱɺ 4616 = 23 × 577 =⇒ v2 (4616) = 3 ͱͳΓɺ͜ͷ਺ ࣗ਎ͷ୅਺తߏ଄ʹΑͬͯɺ௚ޙʹ 3 ࿈ଓͷۮ਺ૢ࡞ ʢ4616 → 2308 → 1154 → 577ʣͱ͍͏ڧྗͳϒϨʔ Ωϋϒ͕֬ఆ͢Δɻ 3.2 ৒༨ྨτϥοϓʹΑΔϋϒ΁ͷೋ࣍ڧ੍࿈ߦ 577ʢ48 ख໨ʣ͸ɺ32 Λ๏ͱͯ͠ 32s+1ʢs = 18ʣ ͱ͍͏ਂ͍৒༨ྨʹ༓ด͞Ε͍ͯΔɻ͜ͷܗͷح਺ ʹରࣸ͠૾Λ࿈ଓద༻͢Δͱɺ୅਺υϛϊʹΑΓ 5 ख ໨ʹۮ਺߲ 27s + 2 ͕ग़ݱ͢Δɻ࣮ࡍͷ஋ s = 18 Λ ୅ೖ͢Δͱ 488 = 23 × 61 =⇒ v2 (488) = 3 ͱͳΓɺ ॳظ஋͕͍࣋ͬͯͨ 32s + 1 ͷෆมੑʹΑΓɺΘ͔ͣ 5 खઌͰ࠶ͼ 23 ͷ 3 ॏϒϨʔΩϋϒ΁׬શಉظͯ͠ ั֫͞ΕΔɻ 3.3 ࢝఺ɾऴ఺ʹ͓͚ΔύϦςΟରশੑͷূ໌ ࠷େఀࢭ࣌ؒͷϨίʔυอ࣋ऀʹ͓͍ͯɺॳظ஋͕ ඞͣʮح਺ʯ ɺશఀࢭ͕࣌ؒඞͣʮۮ਺ʯͱͳΔ਺ཧత ରশੑΛূ໌͢ΔɻୈҰʹɺۮ਺࢝఺͸࠷ॳͷ 1 खͰ x0 /2 ΁ͱڧ੍ॖখ͞Εɺح਺࢝఺͕࣋ͭരൃతͳ৽ ΤωϧΪʔ఺ՐͷݖརΛࣦ͏ͨΊɺϨίʔυ͸ඞͣح ਺͔Βੜ·ΕΔɻୈೋʹɺ࠷ऴऩଋͷखલͷ࠷ޙͷح ਺ϋϒ͸ 5 Ͱ͋Γɺ͔͜͜Β 1 ʹΰʔϧ͢Δ·Ͱͷ ख਺͸Ұҙʹ 5 खʢح਺ʣͰ͋Δɻελʔτ͔Β 5 ʹ ḷΓண͘·Ͱͷख਺͸࿈ཱ߹ಉࣜͷύϦςΟ͔Βඞ ͣح਺ͱͳΔͨΊɺ૯ఀࢭ࣌ؒ͸ ح਺ + ح਺ = ۮ਺ ΁ͱ 100% ऩଋ͢Δɻ 4 يಓ߹ྲྀͷϞδϡϥʔಉظղੳɿ47 ϋϒ ͷղ๤ ਺ཧ࣮ݧͱͯ͠ɺॳظ஋ x0 = 97 ͷ࠷ॳͷ 9 खͷ ڍಈΛධՁ͢Δɻ9 ख໨ͷ౸ୡ਺͸ x9 = 47 Ͱ͋Γɺ ॳظ஋ 27 ͷ 4 ख໨ͷ਺஋ͱ׬શʹಉظʢڧ੍߹ྲྀʣ ͢Δɻ͜ͷͱ͖ͷඍௐ੔߲ S ͸ࣜ (2) ΑΓҎԼͷΑ ͏ʹҰҙʹղ๤͞ΕΔɻ S = 34 ·20 +33 ·20 +32 ·25 +31 ·25 +30 ·20 = 493 (5) 493 ≡ 1 (mod 3) Ͱ͋Γɺิ୊ 1 ͷෆมੑΛຬͨ͢ɻ 47 ͕ଟ͘ͷڊେͳيಓΛҾ͖دͤΔδϟϯΫγϣ ϯ ʢϋϒʣ ͱͳΔཧ༝͸ɺ ٯࣸ૾ T−1(y) = (2y−1)/3 2
3. ͷ੒ཱ৚݅Ͱ͋Δ y ≡ 2 (mod 3) ͷ࿈࠯ʹ͋Δɻ47 ͔Βٯࢉ͞ΕΔ࠯Λ௥੻͢Δͱɺ47 → 94 → 188 → 125 → 83 → 55 ͱͳΓɺ ʮ3 ͰׂΔͱ 2 ༨Δح਺ʯ͕ ਂ͍࿈࠯Λܗ੒͍ͯ͠Δɻ͜ΕʹΑΓɺ্ͷܻͷෆن ଇͳيಓ͸͍ͣΕ͔ͷϊʔυʹ઀৮ͨ͠ॠؒɺ47 ͷ δϟϯΫγϣϯ΁ͱܾఆ࿦తʹڧ੍߹ྲྀͤ͞ΒΕΔɻ 5 LTE ิ୊ʹΑΔϒϨʔΩݕূʢ՝୊ 1 ͷ ಛఆʣ ॳظ஋ x0 ≡ 2k − 1 (mod 2k) ͷ࠷ѱ๫૸ܦ࿏ʹ͓ ͍ͯɺk − 1 ճ࿈ଓͰ͢Γൈ͚ͨ௚ޙͷ k ख໨ʹি ಥ͢Δۮ਺ 3k − 1 ͷ 2 ਐ෇஋ΛධՁ͢ΔͨΊɺLTE ิ୊ʢLifting the Exponent LemmaʣΛಋೖ͢ Δɻ೚ҙͷح਺ k ʹର͢Δ 3k − 1 ͷ 2 ਐ෇஋͸ҎԼ ͷΑ͏ʹݫີʹهड़͞ΕΔɻ v2 (3k−1) = v2 (3−1)+v2 (3+1)+v2 (k)−1 = 1+2+0−1 = 2 (6) ͢ͳΘͪɺฏํͷࠩͷҼ਺෼ղެࣜΛυϛϊ౗͠ʹ ద༻Ͱ͖Δͷ͸ k ͕ 2 ͷྦྷ৐ʢ2, 4, 8, 16...ʣͷͱ ͖ʹݶఆ͞Εɺk ͕ح਺ͷ৔߹ʢྫɿk = 3 ͷͱ͖ 33 − 1 = 26 =⇒ v2 = 1ʣ͸ɺҾ͖໭͠ྗ͕ 2 Ͱ 1 ճׂΔ͚ͩͷ࠷খ஋·Ͱܹݮ͢Δͱ͍͏ੑ࣭͕ଘࡏ ͢Δɻ ͕ͨͬͯ͠ɺ૯ख਺ʹର͢Δۮ਺ૢ࡞ͷൃੜׂ߹ͷ ্ݶʢ36.9% ͷൃࢄݶքൺ཰ [Tao2019]ʣΛ͔͍͙͘ Γɺ͜ͷح਺εςοϓͷ໢ͷ໨ΛӬԕʹ͢Γൈ͚Δݽ ཱܾͨ͠ఆ࿦తൃࢄܦ࿏͕ʮઈରʹଘࡏ͠ͳ͍ʯͱݴ ͍੾ΔͨΊʹ͸ɺ୯ͳΔҼ਺෼ղͷہॴධՁͰ͸ͳ ͘ɺ޿Ҭతͳෆ౳ࣜධՁ͕ඞਢͰ͋Γɺ͜Ε͕ղܾ͢ ΂͖ୈҰͷ՝୊Ͱ͋Δɻ 6 ඇࣗ໌αΠΫϧͷσΟΦϑΝϯτεత՝ ୊ʢ՝୊ 2 ͷಛఆʣ 1 Ҏ֎ͷඇࣗ໌ͳαΠΫϧ͕ଘࡏ͢ΔͨΊͷඞཁ৚ ݅͸ɺࣜ (1) ʹ͓͍ͯ xN = x0 ͱͳΔҎԼͷ୅਺ํ ఔ͕ࣜ x0 > 1 ͷ੔਺ղΛ࣋ͭ͜ͱͰ͋Δɻ x0 = S 2A+B − 3A (7) ϕʔΧʔͷఆཧʢର਺ͷઢܗܗࣜ [Baker1966]ʣΑΓɺ ϧʔϓ੒ཱʹ͸ A + B > 3.5 × 1011ʢ3500 ԯखҎ ্ [Simons2005]ʣ͕ඞཁͰ͋Δɻ͜͜Ͱɺ྆ลΛ 9 Λ๏ͱ͢Δ߹ಉࣜʢ (mod 9)ʣ΁ࣹӨ͢Δͱɺ෼฼ ͷ 2A+B − 3A ͷ৒༨͸ɺݫີͳ 6 ͷपظʢ2, 4, 8, 7, 5, 1ʣʹ׬શʹࢧ഑͞ΕΔɻ͜Εʹର͠ɺ෼ࢠͷ S (mod 9) ͸ɺิ୊ 1 ͷෆมੑʢS ̸≡ 0 (mod 3)ʣΛ࣋ ͪͭͭ΋ɺෆنଇͳ 3 ਐσδλϧͷ೾Λܗ੒͢Δɻ ͜ͷ௒ڊେ਺ͷੈքʹ͓͍ͯɺ෼฼ͷඳ͘୯७ͳ 2 ਐͷ 6 पظͷ೾ͱɺ෼ࢠͷඳ͘ 3 ਐσδλϧͷෆنଇ ͳ೾͕ɺࣗ໌ͳղ x0 = 1 Λআ͍ͯʮઈରʹҰக͠ͳ ͍ʢׂΓ੾Εͳ͍ʣ ʯͱݴ͍੾ΔͨΊʹ͸ɺ෼ࢠ S ͷ ࣋ͭૉҼ਺ͷू߹͕෼฼ 2A+B − 3A ͷ࣋ͭૉҼ਺ͷ ू߹Λ׬શʹแઁʢؙವΈʣͰ͖ͳ͍͜ͱΛɺp ਐ੔ ਺؀ Zp ্ͷ௒ӽ਺࿦తෆ౳ࣜͱͯ͠ݫີʹ͖ࣔ͠Δ ඞཁ͕͋Γɺ͜Ε͕ղܾ͢΂͖ୈೋͷ՝୊Ͱ͋Δɻ 7 ݁࿦ ຊݚڀʹΑΓɺෆنଇيಓΛࢧ഑͢ΔҰൠ͔ࣜΒɺ ඍௐ੔߲ͷϞδϡϥʔෆมੑɺ2 ਐ෇஋ʹΑΔϒϨʔ Ωɺೖग़ྗͷύϦςΟରশੑɺ47 ϋϒͷ஖ᥨͷ૥ߏ ଄ɺ͓ΑͼൃࢄͱαΠΫϧͷଘཱݶքઢ͕Ұ੾ͷޡຐ Խ͠ͳ͘਺ࣜԽ͞Εͨɻίϥοπ༧૝ͷ׬શղܾͷ ͨΊʹ͸ɺ্ड़ͨ͠ʮ՝୊ 1ɿح਺εςοϓʹ͓͚Δ Ҿ͖໭͠ྗܹݮͷධՁʯ͓Αͼʮ՝୊ 2ɿ௒ڊେ਺ʹ ͓͚Δ෼ࢠ෼฼ͷ p ਐτϙϩδʔతඇಉظͷ׬શه ड़ʯͷ 2 ఺Λɺ୅਺زԿֶ͓Αͼ௒ӽ਺࿦ͷ࿮૊ΈΛ ༻͍ͯ׬શʹด࠹͢Δඞཁ͕͋Γɺ͜ΕΛࠓޙͷݚڀ ՝୊ͱ͢Δɻ ࢀߟจݙ [1] T. Tao, “Almost all orbits of the Collatz map attain almost bounded values,” arXiv preprint arXiv:1909.03562, 2019. [2] J. L. Simons and B. M. M. de Weger, “Theo- retical and computational bounds for m-cycles of the 3x + 1 problem,” Acta Arithmetica, vol. 117, no. 1, pp. 51–70, 2005. [3] A. Baker, “Linear forms in the logarithms of algebraic numbers,” Mathematika, vol. 13, no. 2, pp. 204–216, 1966. [4] J. C. Lagarias, “The 3x + 1 problem and its generalizations,” The American Mathematical Monthly, vol. 92, no. 1, pp. 3–23, 1985. 3