# Registre K : paliers 2-adique (m=11 à m=16) ## Introduction Ce document trace, de manière reproductible, l’évolution d’un registre de clauses universelles (D et F) appliquées à la dynamique U(n) (impairs → impairs). À chaque palier m, l’ensemble des résidus impairs modulo 2^m est partitionné en classes couvertes (au sens : une clause du registre s’applique de façon universelle au-delà d’un seuil) et un résidu non couvert. Les listes de résidus non couverts sont exhaustives. ## Définitions Pour n impair : - a(n) = v2(3n+1) (valuation 2-adique). - U(n) = (3n+1) / 2^{a(n)} (retourne un impair). Une clause de descente (D) utilisée ici est dérivée d’un bloc de valuations exactes (a0,…,a_{k-1}) de somme A_k et du terme additif C_k : - A_0 = 0 ; A_{i+1} = A_i + a_i. - C_0 = 0 ; C_{i+1} = 3*C_i + 2^{A_i}. - U^{(k)}(n) = (3^k * n + C_k) / 2^{A_k}. - Δ_k = 2^{A_k} − 3^k ; N0 = floor(C_k/Δ_k) + 1. Une clause de fusion (F) utilisée ici est de type préimage courte a=1 : - si y ≡ 5 (mod 6), alors m = (2y−1)/3 est impair, m < y, et U(m)=y. - appliquée à y = U^{(t)}(n) stable sur une classe. ## Règles de base incluses (clauses grossières) - V : si n ≡ 1 (mod 4) et n ≥ 3, alors U(n) < n. - D2 : si n ≡ 3 (mod 16), alors U^2(n) < n (borne sur la valuation au 2e pas). - D3 : si n ≡ 11 (mod 32) ou n ≡ 23 (mod 32), alors U^3(n) < n (borne sur la valuation au 3e pas). ## Critère de stabilité au palier 2^m Une clause issue d’un bloc exact de somme A est considérée applicable à un résidu r (mod 2^m) si A + 1 ≤ m (car alors r (mod 2^m) ⇒ r (mod 2^{A+1}) et le bloc est stable sur la classe). ## Résultats synthétiques | m | 2^m | impairs | couverts | non couverts | taux de couverture | |---:|---:|---:|---:|---:|---:| | 11 | 2048 | 1024 | 890 | 134 | 0.869140625000 | | 12 | 4096 | 2048 | 1812 | 236 | 0.884765625000 | | 13 | 8192 | 4096 | 3668 | 428 | 0.895507812500 | | 14 | 16384 | 8192 | 7440 | 752 | 0.908203125000 | | 15 | 32768 | 16384 | 15039 | 1345 | 0.917907714844 | | 16 | 65536 | 32768 | 30322 | 2446 | 0.925354003906 | ## Listes exhaustives des résidus non couverts ### Palier m = 11 (mod 2^11 = 2048) Nombre de résidus impairs non couverts : 134 sur 1024. 27, 31, 47, 63, 71, 91, 103, 111, 127, 155, 159, 167, 191, 223, 231, 239 251, 255, 283, 303, 319, 327, 347, 359, 367, 383, 415, 423, 447, 479, 495, 507 511, 539, 543, 559, 575, 583, 603, 615, 623, 639, 667, 671, 679, 703, 735, 743 751, 763, 767, 783, 795, 831, 839, 859, 871, 895, 923, 927, 935, 943, 959, 991 999, 1007, 1019, 1023, 1051, 1055, 1071, 1087, 1095, 1115, 1127, 1135, 1151, 1179, 1183, 1191 1215, 1247, 1255, 1263, 1275, 1279, 1307, 1311, 1327, 1339, 1343, 1351, 1383, 1407, 1415, 1439 1471, 1499, 1503, 1519, 1535, 1563, 1567, 1575, 1583, 1627, 1639, 1647, 1659, 1663, 1691, 1695 1703, 1727, 1735, 1767, 1775, 1787, 1791, 1819, 1839, 1855, 1863, 1883, 1887, 1895, 1919, 1951 1959, 1983, 2015, 2031, 2043, 2047 ### Palier m = 12 (mod 2^12 = 4096) Nombre de résidus impairs non couverts : 236 sur 2048. 27, 31, 47, 63, 71, 91, 103, 111, 127, 159, 167, 191, 223, 231, 239, 251 255, 283, 303, 319, 327, 347, 359, 383, 415, 447, 479, 495, 511, 539, 559, 583 603, 615, 623, 639, 667, 671, 679, 703, 735, 743, 751, 763, 767, 795, 831, 839 859, 871, 895, 927, 935, 943, 959, 991, 999, 1007, 1019, 1023, 1051, 1055, 1087, 1095 1115, 1127, 1135, 1151, 1179, 1183, 1191, 1215, 1247, 1255, 1263, 1275, 1279, 1307, 1327, 1343 1351, 1383, 1407, 1415, 1439, 1471, 1499, 1503, 1519, 1535, 1563, 1567, 1583, 1639, 1647, 1663 1691, 1695, 1703, 1727, 1735, 1767, 1775, 1787, 1791, 1819, 1855, 1883, 1887, 1895, 1919, 1951 1959, 1983, 2031, 2043, 2047, 2075, 2079, 2095, 2111, 2119, 2139, 2151, 2159, 2175, 2203, 2207 2215, 2239, 2271, 2287, 2299, 2303, 2331, 2351, 2367, 2375, 2407, 2415, 2463, 2471, 2495, 2527 2543, 2555, 2559, 2587, 2591, 2607, 2623, 2651, 2663, 2671, 2687, 2715, 2719, 2727, 2751, 2791 2799, 2811, 2815, 2831, 2843, 2879, 2887, 2907, 2919, 2943, 2971, 2975, 2983, 3007, 3039, 3055 3067, 3071, 3099, 3103, 3119, 3135, 3143, 3163, 3175, 3183, 3199, 3227, 3231, 3239, 3263, 3295 3303, 3311, 3323, 3327, 3355, 3359, 3375, 3387, 3391, 3399, 3431, 3455, 3487, 3519, 3551, 3567 3583, 3611, 3615, 3623, 3631, 3675, 3687, 3707, 3711, 3739, 3743, 3775, 3815, 3823, 3839, 3867 3887, 3911, 3931, 3943, 3967, 3999, 4007, 4031, 4063, 4079, 4091, 4095 ### Nouveaux résidus couverts en passant de m = 11 à m = 12 Nombre : 32 (parmi les deux enfants modulo 2^12 des résidus non couverts au palier précédent). 155, 367, 423, 507, 543, 575, 783, 923, 1071, 1311, 1339, 1575, 1627, 1659, 1839, 1863 2015, 2279, 2395, 2431, 2631, 2783, 2991, 3047, 3463, 3547, 3695, 3751, 3783, 3835, 3903, 3935 Annotations (type, horizon, somme A, seuil N) pour les 24 premiers : - r=155 : F ; horizon=7 ; A=11 ; N=3 - r=367 : D ; horizon=6 ; A=11 ; N=1 - r=423 : D ; horizon=6 ; A=11 ; N=1 - r=507 : D ; horizon=6 ; A=11 ; N=1 - r=543 : F ; horizon=7 ; A=11 ; N=2 - r=575 : D ; horizon=6 ; A=11 ; N=1 - r=783 : D ; horizon=4 ; A=11 ; N=1 - r=923 : D ; horizon=6 ; A=11 ; N=1 - r=1071 : F ; horizon=7 ; A=11 ; N=2 - r=1311 : D ; horizon=6 ; A=11 ; N=1 - r=1339 : D ; horizon=4 ; A=11 ; N=1 - r=1575 : D ; horizon=5 ; A=11 ; N=1 - r=1627 : F ; horizon=7 ; A=11 ; N=3 - r=1659 : D ; horizon=5 ; A=11 ; N=1 - r=1839 : D ; horizon=6 ; A=11 ; N=1 - r=1863 : F ; horizon=7 ; A=11 ; N=3 - r=2015 : F ; horizon=7 ; A=11 ; N=2 - r=2279 : F ; horizon=7 ; A=11 ; N=2 - r=2395 : D ; horizon=6 ; A=11 ; N=1 - r=2431 : F ; horizon=7 ; A=11 ; N=2 - r=2631 : D ; horizon=6 ; A=11 ; N=1 - r=2783 : D ; horizon=6 ; A=11 ; N=1 - r=2991 : D ; horizon=5 ; A=11 ; N=1 - r=3047 : D ; horizon=6 ; A=11 ; N=1 ### Palier m = 13 (mod 2^13 = 8192) Nombre de résidus impairs non couverts : 428 sur 4096. 27, 31, 47, 63, 71, 91, 103, 111, 127, 159, 167, 191, 223, 231, 239, 251 255, 283, 303, 319, 327, 347, 359, 415, 447, 479, 495, 511, 539, 559, 583, 603 623, 639, 667, 671, 679, 703, 743, 751, 763, 767, 795, 831, 839, 859, 871, 895 927, 935, 943, 959, 991, 999, 1007, 1023, 1051, 1055, 1095, 1115, 1127, 1135, 1151, 1179 1183, 1191, 1215, 1247, 1255, 1263, 1275, 1279, 1307, 1327, 1343, 1351, 1383, 1407, 1415, 1439 1471, 1503, 1519, 1535, 1563, 1567, 1583, 1639, 1647, 1663, 1691, 1695, 1703, 1727, 1767, 1775 1791, 1819, 1883, 1887, 1895, 1919, 1951, 1959, 1983, 2043, 2047, 2075, 2079, 2095, 2111, 2119 2139, 2151, 2159, 2175, 2207, 2215, 2239, 2271, 2287, 2299, 2303, 2331, 2351, 2367, 2375, 2407 2415, 2463, 2471, 2495, 2527, 2543, 2559, 2587, 2591, 2607, 2651, 2663, 2671, 2687, 2715, 2719 2727, 2751, 2791, 2799, 2811, 2815, 2831, 2843, 2879, 2887, 2919, 2943, 2983, 3007, 3039, 3055 3067, 3071, 3099, 3103, 3119, 3135, 3163, 3175, 3183, 3199, 3227, 3231, 3239, 3263, 3295, 3303 3311, 3323, 3327, 3355, 3359, 3375, 3391, 3399, 3431, 3455, 3487, 3519, 3551, 3567, 3583, 3611 3615, 3631, 3687, 3707, 3711, 3739, 3743, 3775, 3815, 3823, 3839, 3867, 3887, 3931, 3943, 3967 3999, 4007, 4031, 4063, 4079, 4091, 4095, 4123, 4127, 4143, 4159, 4167, 4187, 4199, 4207, 4223 4255, 4263, 4287, 4319, 4335, 4347, 4351, 4379, 4399, 4415, 4423, 4455, 4479, 4511, 4543, 4575 4591, 4607, 4635, 4655, 4699, 4711, 4719, 4735, 4763, 4767, 4775, 4799, 4831, 4839, 4847, 4859 4863, 4891, 4927, 4935, 4955, 4967, 4991, 5023, 5055, 5087, 5103, 5115, 5119, 5147, 5151, 5183 5191, 5211, 5223, 5231, 5247, 5275, 5279, 5287, 5311, 5343, 5351, 5359, 5371, 5375, 5403, 5423 5439, 5447, 5479, 5503, 5535, 5567, 5595, 5599, 5615, 5631, 5659, 5663, 5679, 5735, 5759, 5787 5791, 5823, 5831, 5863, 5871, 5883, 5887, 5915, 5951, 5979, 5991, 6015, 6047, 6055, 6079, 6127 6139, 6143, 6171, 6175, 6191, 6207, 6215, 6235, 6247, 6255, 6271, 6299, 6303, 6311, 6367, 6383 6395, 6399, 6427, 6463, 6471, 6503, 6559, 6591, 6623, 6639, 6651, 6655, 6703, 6719, 6747, 6759 6767, 6783, 6811, 6815, 6823, 6847, 6887, 6895, 6907, 6911, 6939, 6975, 6983, 7003, 7015, 7039 7067, 7071, 7079, 7103, 7135, 7151, 7163, 7167, 7195, 7199, 7231, 7239, 7259, 7271, 7279, 7295 7323, 7327, 7335, 7359, 7399, 7407, 7419, 7423, 7451, 7471, 7483, 7487, 7495, 7527, 7551, 7583 7615, 7647, 7663, 7679, 7707, 7711, 7719, 7727, 7771, 7783, 7807, 7835, 7839, 7871, 7911, 7919 7935, 7963, 8007, 8027, 8039, 8063, 8095, 8103, 8127, 8175, 8187, 8191 ### Nouveaux résidus couverts en passant de m = 12 à m = 13 Nombre : 44 (parmi les deux enfants modulo 2^13 des résidus non couverts au palier précédent). 383, 615, 735, 1019, 1087, 1499, 1735, 1787, 1855, 2031, 2203, 2555, 2623, 2907, 2971, 2975 3143, 3387, 3623, 3675, 3911, 4327, 4443, 4679, 5031, 5039, 5095, 5511, 5743, 5799, 5983, 6335 6447, 6511, 6567, 6683, 6687, 6927, 7215, 7391, 7455, 7803, 7983, 8159 Annotations (type, horizon, somme A, seuil N) pour les 24 premiers : - r=383 : D ; horizon=7 ; A=12 ; N=2 - r=615 : D ; horizon=7 ; A=12 ; N=2 - r=735 : D ; horizon=6 ; A=12 ; N=1 - r=1019 : D ; horizon=7 ; A=12 ; N=2 - r=1087 : D ; horizon=7 ; A=12 ; N=2 - r=1499 : D ; horizon=5 ; A=12 ; N=1 - r=1735 : D ; horizon=5 ; A=12 ; N=1 - r=1787 : D ; horizon=7 ; A=12 ; N=2 - r=1855 : D ; horizon=7 ; A=12 ; N=2 - r=2031 : D ; horizon=7 ; A=12 ; N=2 - r=2203 : D ; horizon=7 ; A=12 ; N=2 - r=2555 : D ; horizon=6 ; A=12 ; N=1 - r=2623 : D ; horizon=6 ; A=12 ; N=1 - r=2907 : D ; horizon=7 ; A=12 ; N=2 - r=2971 : D ; horizon=6 ; A=12 ; N=1 - r=2975 : D ; horizon=7 ; A=12 ; N=2 - r=3143 : D ; horizon=7 ; A=12 ; N=2 - r=3387 : D ; horizon=4 ; A=12 ; N=1 - r=3623 : D ; horizon=5 ; A=12 ; N=1 - r=3675 : D ; horizon=7 ; A=12 ; N=2 - r=3911 : D ; horizon=7 ; A=12 ; N=2 - r=4327 : D ; horizon=7 ; A=12 ; N=2 - r=4443 : D ; horizon=6 ; A=12 ; N=1 - r=4679 : D ; horizon=6 ; A=12 ; N=1 ### Palier m = 14 (mod 2^14 = 16384) Nombre de résidus impairs non couverts : 752 sur 8192. 27, 31, 47, 63, 71, 91, 103, 111, 127, 159, 167, 191, 223, 231, 239, 251 283, 303, 319, 327, 359, 415, 447, 479, 495, 511, 559, 583, 603, 623, 639, 667 671, 703, 743, 751, 763, 767, 795, 831, 839, 859, 871, 895, 927, 935, 943, 959 991, 1007, 1023, 1051, 1055, 1095, 1115, 1127, 1135, 1151, 1179, 1183, 1215, 1247, 1255, 1263 1275, 1279, 1307, 1327, 1343, 1351, 1383, 1407, 1415, 1439, 1471, 1503, 1519, 1535, 1567, 1583 1639, 1647, 1663, 1691, 1695, 1727, 1767, 1775, 1791, 1819, 1883, 1887, 1895, 1919, 1951, 1959 2043, 2047, 2079, 2111, 2119, 2139, 2151, 2159, 2175, 2207, 2215, 2239, 2271, 2287, 2299, 2303 2331, 2351, 2367, 2375, 2407, 2463, 2471, 2495, 2527, 2543, 2559, 2591, 2651, 2663, 2671, 2687 2715, 2719, 2727, 2751, 2791, 2799, 2811, 2815, 2831, 2843, 2879, 2887, 2919, 2943, 2983, 3007 3055, 3067, 3071, 3099, 3103, 3135, 3163, 3175, 3183, 3199, 3227, 3231, 3239, 3263, 3295, 3303 3311, 3323, 3327, 3355, 3375, 3391, 3399, 3431, 3455, 3487, 3519, 3567, 3583, 3611, 3615, 3631 3687, 3707, 3711, 3739, 3743, 3775, 3815, 3823, 3839, 3867, 3887, 3931, 3943, 3967, 3999, 4007 4031, 4079, 4091, 4095, 4123, 4127, 4143, 4159, 4167, 4187, 4199, 4207, 4255, 4263, 4287, 4319 4335, 4347, 4351, 4379, 4399, 4415, 4423, 4479, 4511, 4543, 4575, 4591, 4607, 4635, 4655, 4699 4711, 4719, 4735, 4763, 4767, 4775, 4799, 4831, 4839, 4847, 4863, 4891, 4935, 4955, 4967, 4991 5023, 5055, 5087, 5103, 5115, 5119, 5147, 5151, 5183, 5211, 5223, 5231, 5247, 5275, 5279, 5287 5311, 5343, 5351, 5359, 5375, 5403, 5423, 5447, 5479, 5503, 5535, 5567, 5595, 5599, 5615, 5631 5659, 5663, 5679, 5735, 5759, 5787, 5791, 5823, 5863, 5887, 5915, 5979, 5991, 6015, 6047, 6055 6079, 6127, 6139, 6143, 6171, 6175, 6191, 6207, 6235, 6247, 6255, 6271, 6299, 6303, 6311, 6367 6383, 6395, 6399, 6427, 6463, 6471, 6503, 6559, 6591, 6623, 6639, 6651, 6655, 6703, 6719, 6759 6767, 6783, 6811, 6823, 6847, 6887, 6895, 6907, 6911, 6939, 6975, 6983, 7003, 7015, 7039, 7071 7103, 7135, 7151, 7163, 7167, 7195, 7199, 7231, 7271, 7279, 7295, 7323, 7327, 7335, 7359, 7407 7419, 7423, 7451, 7471, 7483, 7487, 7495, 7527, 7551, 7583, 7615, 7647, 7663, 7679, 7707, 7711 7727, 7783, 7807, 7835, 7839, 7871, 7919, 7935, 7963, 8007, 8027, 8039, 8063, 8095, 8127, 8175 8187, 8191, 8219, 8223, 8239, 8255, 8263, 8283, 8295, 8303, 8319, 8351, 8359, 8415, 8431, 8443 8447, 8475, 8511, 8519, 8539, 8551, 8607, 8639, 8671, 8687, 8703, 8731, 8751, 8795, 8831, 8859 8863, 8871, 8895, 8935, 8943, 8955, 8959, 8987, 9023, 9031, 9051, 9063, 9087, 9119, 9151, 9183 9191, 9199, 9215, 9243, 9247, 9287, 9307, 9319, 9343, 9371, 9375, 9383, 9447, 9455, 9467, 9471 9499, 9535, 9543, 9575, 9599, 9631, 9663, 9695, 9711, 9727, 9755, 9775, 9831, 9855, 9883, 9887 9895, 9959, 9967, 9983, 10011, 10075, 10087, 10111, 10143, 10151, 10175, 10235, 10239, 10267, 10287, 10303 10311, 10331, 10343, 10351, 10367, 10399, 10407, 10479, 10491, 10495, 10559, 10567, 10599, 10607, 10655, 10687 10719, 10735, 10751, 10779, 10799, 10843, 10863, 10879, 10907, 10911, 10919, 10943, 10983, 10991, 11003, 11007 11035, 11071, 11079, 11111, 11135, 11175, 11199, 11231, 11247, 11263, 11291, 11295, 11311, 11355, 11367, 11375 11391, 11419, 11423, 11431, 11455, 11495, 11503, 11515, 11519, 11547, 11551, 11567, 11583, 11591, 11623, 11679 11711, 11743, 11759, 11775, 11803, 11807, 11823, 11903, 11931, 11935, 11967, 12007, 12015, 12031, 12059, 12123 12135, 12191, 12199, 12223, 12255, 12287, 12315, 12319, 12335, 12359, 12379, 12399, 12415, 12447, 12455, 12479 12511, 12527, 12539, 12543, 12571, 12591, 12607, 12615, 12647, 12703, 12735, 12767, 12783, 12799, 12827, 12847 12891, 12911, 12927, 12955, 12959, 12967, 12991, 13031, 13039, 13051, 13055, 13083, 13119, 13127, 13159, 13183 13247, 13279, 13311, 13339, 13343, 13383, 13403, 13415, 13423, 13439, 13471, 13479, 13503, 13535, 13543, 13551 13563, 13567, 13595, 13615, 13631, 13639, 13671, 13695, 13727, 13759, 13791, 13823, 13851, 13855, 13871, 13927 13951, 13983, 14015, 14023, 14055, 14063, 14075, 14079, 14107, 14143, 14183, 14207, 14247, 14271, 14331, 14335 14363, 14367, 14383, 14399, 14407, 14427, 14439, 14447, 14463, 14495, 14503, 14559, 14575, 14587, 14591, 14619 14655, 14663, 14695, 14783, 14815, 14831, 14847, 14895, 14939, 14951, 14959, 14975, 15003, 15007, 15015, 15039 15079, 15087, 15099, 15103, 15131, 15167, 15207, 15231, 15259, 15271, 15295, 15327, 15343, 15355, 15359, 15387 15391, 15423, 15431, 15451, 15463, 15471, 15487, 15515, 15519, 15527, 15551, 15591, 15599, 15611, 15615, 15643 15663, 15679, 15719, 15743, 15775, 15807, 15839, 15855, 15871, 15899, 15903, 15911, 15919, 15963, 15975, 15999 16027, 16031, 16063, 16103, 16111, 16127, 16155, 16219, 16231, 16255, 16287, 16295, 16319, 16367, 16379, 16383 ### Nouveaux résidus couverts en passant de m = 13 à m = 14 Nombre : 104 (parmi les deux enfants modulo 2^14 des résidus non couverts au palier précédent). 255, 347, 539, 679, 999, 1191, 1563, 1703, 1983, 2075, 2095, 2415, 2587, 2607, 3039, 3119 3359, 3551, 4063, 4223, 4455, 4859, 4927, 5191, 5371, 5439, 5831, 5871, 5883, 5951, 6215, 6747 6815, 7067, 7079, 7239, 7259, 7399, 7719, 7771, 7911, 8103, 8383, 8423, 8495, 8775, 8815, 9127 9135, 9327, 9407, 9439, 9519, 9607, 9759, 9839, 9919, 10079, 10271, 10431, 10463, 10523, 10543, 10663 10783, 10855, 11023, 11259, 11327, 11487, 11647, 11879, 11899, 12079, 12159, 12271, 12283, 12351, 12391, 12671 12903, 13023, 13147, 13215, 13295, 13307, 13375, 13467, 13787, 13807, 13979, 14171, 14239, 14319, 14491, 14751 14843, 14911, 15175, 15195, 15263, 15675, 15687, 16199 Annotations (type, horizon, somme A, seuil N) pour les 24 premiers : - r=255 : D ; horizon=8 ; A=13 ; N=4 - r=347 : D ; horizon=6 ; A=13 ; N=1 - r=539 : D ; horizon=8 ; A=13 ; N=7 - r=679 : D ; horizon=8 ; A=13 ; N=6 - r=999 : D ; horizon=6 ; A=13 ; N=1 - r=1191 : D ; horizon=8 ; A=13 ; N=6 - r=1563 : D ; horizon=8 ; A=13 ; N=6 - r=1703 : D ; horizon=7 ; A=13 ; N=1 - r=1983 : D ; horizon=8 ; A=13 ; N=5 - r=2075 : D ; horizon=8 ; A=13 ; N=6 - r=2095 : D ; horizon=8 ; A=13 ; N=6 - r=2415 : D ; horizon=6 ; A=13 ; N=1 - r=2587 : D ; horizon=7 ; A=13 ; N=1 - r=2607 : D ; horizon=8 ; A=13 ; N=6 - r=3039 : D ; horizon=8 ; A=13 ; N=6 - r=3119 : D ; horizon=7 ; A=13 ; N=1 - r=3359 : D ; horizon=6 ; A=13 ; N=1 - r=3551 : D ; horizon=8 ; A=13 ; N=5 - r=4063 : D ; horizon=7 ; A=13 ; N=1 - r=4223 : D ; horizon=8 ; A=13 ; N=4 - r=4455 : D ; horizon=8 ; A=13 ; N=5 - r=4859 : D ; horizon=8 ; A=13 ; N=8 - r=4927 : D ; horizon=8 ; A=13 ; N=5 - r=5191 : D ; horizon=8 ; A=13 ; N=8 ### Palier m = 15 (mod 2^15 = 32768) Nombre de résidus impairs non couverts : 1345 sur 16384. 27, 31, 47, 63, 71, 91, 103, 111, 127, 159, 167, 223, 239, 251, 283, 303 319, 327, 359, 415, 447, 479, 495, 511, 559, 583, 603, 623, 639, 667, 671, 703 743, 751, 763, 767, 795, 831, 839, 859, 871, 895, 927, 943, 959, 991, 1007, 1023 1051, 1055, 1095, 1115, 1127, 1151, 1179, 1183, 1247, 1255, 1263, 1275, 1279, 1307, 1327, 1343 1383, 1407, 1439, 1471, 1503, 1519, 1535, 1567, 1583, 1639, 1647, 1663, 1691, 1695, 1727, 1767 1775, 1791, 1819, 1883, 1887, 1895, 1919, 1951, 1959, 2043, 2047, 2111, 2119, 2139, 2151, 2159 2175, 2207, 2215, 2271, 2287, 2299, 2303, 2331, 2351, 2367, 2375, 2407, 2463, 2471, 2495, 2527 2543, 2559, 2591, 2651, 2671, 2687, 2715, 2719, 2727, 2751, 2791, 2799, 2811, 2831, 2843, 2879 2887, 2919, 2943, 2983, 3007, 3055, 3071, 3099, 3103, 3135, 3163, 3175, 3183, 3199, 3227, 3231 3263, 3295, 3303, 3311, 3323, 3327, 3355, 3375, 3391, 3399, 3431, 3455, 3487, 3519, 3567, 3583 3611, 3615, 3631, 3711, 3739, 3743, 3775, 3815, 3823, 3839, 3867, 3931, 3943, 3999, 4007, 4031 4079, 4095, 4127, 4159, 4167, 4187, 4199, 4207, 4255, 4263, 4287, 4319, 4335, 4347, 4351, 4379 4399, 4415, 4423, 4479, 4511, 4575, 4591, 4607, 4635, 4655, 4699, 4719, 4735, 4763, 4767, 4775 4799, 4839, 4847, 4863, 4891, 4935, 4967, 4991, 5023, 5055, 5103, 5119, 5147, 5151, 5183, 5211 5223, 5231, 5247, 5279, 5287, 5311, 5343, 5351, 5359, 5375, 5403, 5423, 5447, 5479, 5503, 5535 5567, 5599, 5631, 5659, 5663, 5679, 5735, 5759, 5787, 5791, 5823, 5863, 5887, 5915, 5991, 6015 6047, 6055, 6079, 6127, 6139, 6143, 6171, 6175, 6191, 6207, 6235, 6247, 6255, 6303, 6311, 6367 6383, 6395, 6399, 6427, 6463, 6471, 6503, 6591, 6623, 6639, 6651, 6655, 6703, 6759, 6767, 6783 6811, 6823, 6847, 6887, 6895, 6907, 6911, 6939, 7039, 7071, 7103, 7135, 7151, 7163, 7167, 7195 7199, 7231, 7271, 7279, 7295, 7323, 7327, 7335, 7359, 7407, 7423, 7451, 7471, 7487, 7495, 7527 7551, 7583, 7615, 7647, 7663, 7679, 7707, 7711, 7727, 7783, 7807, 7835, 7839, 7871, 7935, 7963 8027, 8039, 8063, 8095, 8127, 8175, 8187, 8191, 8219, 8223, 8239, 8255, 8283, 8295, 8303, 8319 8351, 8359, 8415, 8431, 8443, 8475, 8511, 8519, 8539, 8551, 8607, 8639, 8671, 8687, 8703, 8751 8795, 8831, 8859, 8895, 8935, 8943, 8955, 8959, 8987, 9023, 9031, 9051, 9063, 9087, 9119, 9151 9183, 9199, 9215, 9243, 9247, 9287, 9319, 9343, 9371, 9375, 9383, 9455, 9467, 9471, 9499, 9535 9543, 9575, 9599, 9631, 9663, 9695, 9711, 9727, 9775, 9831, 9855, 9883, 9887, 9959, 9967, 9983 10011, 10075, 10087, 10111, 10143, 10235, 10239, 10267, 10287, 10303, 10311, 10331, 10343, 10351, 10367, 10399 10407, 10479, 10491, 10495, 10559, 10567, 10599, 10607, 10655, 10687, 10719, 10735, 10751, 10843, 10863, 10879 10907, 10911, 10919, 10943, 10983, 10991, 11003, 11007, 11035, 11071, 11079, 11111, 11135, 11175, 11199, 11231 11247, 11263, 11291, 11295, 11311, 11355, 11367, 11391, 11419, 11423, 11431, 11495, 11503, 11515, 11519, 11547 11551, 11567, 11583, 11591, 11623, 11679, 11711, 11759, 11775, 11803, 11807, 11823, 11903, 11931, 11935, 11967 12007, 12015, 12031, 12059, 12123, 12135, 12191, 12199, 12223, 12255, 12287, 12315, 12335, 12359, 12379, 12399 12415, 12447, 12455, 12479, 12511, 12527, 12539, 12543, 12571, 12591, 12607, 12615, 12703, 12735, 12767, 12783 12799, 12827, 12847, 12891, 12911, 12927, 12955, 12959, 12967, 12991, 13031, 13039, 13055, 13119, 13127, 13159 13183, 13247, 13279, 13311, 13339, 13343, 13383, 13403, 13415, 13423, 13439, 13471, 13479, 13503, 13535, 13543 13551, 13563, 13567, 13595, 13615, 13639, 13671, 13695, 13727, 13759, 13791, 13823, 13851, 13855, 13871, 13951 13983, 14015, 14023, 14055, 14063, 14079, 14107, 14143, 14183, 14247, 14271, 14335, 14363, 14367, 14383, 14399 14407, 14427, 14439, 14447, 14463, 14495, 14503, 14559, 14575, 14587, 14591, 14619, 14655, 14663, 14695, 14783 14815, 14831, 14847, 14895, 14951, 14959, 14975, 15003, 15007, 15015, 15039, 15079, 15087, 15099, 15103, 15131 15167, 15207, 15231, 15271, 15295, 15327, 15343, 15355, 15359, 15387, 15391, 15423, 15431, 15451, 15463, 15471 15487, 15519, 15527, 15551, 15591, 15599, 15611, 15615, 15643, 15663, 15679, 15719, 15743, 15775, 15807, 15839 15871, 15899, 15903, 15911, 15919, 15975, 15999, 16027, 16031, 16063, 16111, 16127, 16155, 16231, 16255, 16287 16295, 16319, 16367, 16379, 16383, 16411, 16415, 16431, 16447, 16455, 16475, 16487, 16495, 16511, 16543, 16551 16575, 16607, 16615, 16623, 16635, 16667, 16703, 16711, 16743, 16831, 16863, 16879, 16895, 16943, 16987, 17023 17051, 17055, 17087, 17127, 17135, 17147, 17151, 17179, 17215, 17243, 17255, 17279, 17311, 17319, 17343, 17375 17391, 17407, 17435, 17439, 17479, 17499, 17511, 17519, 17535, 17563, 17567, 17599, 17639, 17647, 17659, 17663 17691, 17727, 17735, 17767, 17791, 17799, 17823, 17855, 17887, 17903, 17919, 17967, 18023, 18047, 18075, 18079 18151, 18159, 18175, 18203, 18267, 18279, 18303, 18335, 18343, 18427, 18431, 18463, 18495, 18503, 18523, 18535 18543, 18559, 18591, 18599, 18623, 18671, 18683, 18751, 18759, 18791, 18847, 18879, 18911, 18927, 18943, 19035 19047, 19055, 19071, 19099, 19103, 19135, 19175, 19183, 19195, 19199, 19227, 19263, 19271, 19303, 19327, 19367 19391, 19439, 19451, 19455, 19483, 19487, 19547, 19559, 19567, 19583, 19611, 19615, 19623, 19647, 19687, 19695 19707, 19711, 19739, 19759, 19775, 19783, 19815, 19871, 19903, 19951, 19967, 19999, 20015, 20071, 20091, 20095 20123, 20127, 20159, 20199, 20207, 20223, 20251, 20271, 20315, 20327, 20351, 20383, 20391, 20475, 20479, 20507 20511, 20527, 20551, 20571, 20591, 20639, 20647, 20671, 20703, 20719, 20731, 20735, 20763, 20783, 20799, 20807 20895, 20927, 20959, 20975, 20991, 21019, 21083, 21095, 21103, 21119, 21147, 21151, 21159, 21183, 21215, 21223 21231, 21247, 21275, 21319, 21339, 21351, 21375, 21439, 21471, 21499, 21503, 21531, 21535, 21595, 21607, 21615 21631, 21659, 21663, 21671, 21695, 21727, 21735, 21743, 21759, 21787, 21807, 21831, 21863, 21887, 21919, 21951 21979, 21999, 22015, 22043, 22047, 22063, 22119, 22143, 22175, 22207, 22247, 22271, 22299, 22363, 22375, 22399 22439, 22463, 22523, 22527, 22555, 22559, 22575, 22591, 22619, 22631, 22639, 22655, 22683, 22687, 22695, 22751 22767, 22779, 22783, 22811, 22847, 22855, 22943, 22975, 23007, 23023, 23039, 23087, 23103, 23143, 23151, 23195 23207, 23231, 23271, 23279, 23295, 23323, 23359, 23367, 23387, 23399, 23423, 23487, 23519, 23535, 23547, 23551 23579, 23583, 23615, 23655, 23663, 23679, 23707, 23711, 23719, 23743, 23791, 23803, 23807, 23835, 23855, 23867 23911, 23935, 23967, 23999, 24031, 24047, 24063, 24091, 24095, 24111, 24167, 24191, 24219, 24223, 24255, 24303 24319, 24347, 24391, 24411, 24423, 24447, 24479, 24511, 24559, 24571, 24575, 24603, 24607, 24623, 24639, 24647 24667, 24679, 24687, 24703, 24735, 24743, 24799, 24827, 24831, 24859, 24895, 24903, 24935, 24991, 25023, 25055 25071, 25087, 25115, 25135, 25215, 25243, 25247, 25255, 25279, 25319, 25327, 25339, 25343, 25371, 25407, 25415 25435, 25447, 25471, 25503, 25535, 25567, 25575, 25583, 25599, 25627, 25631, 25671, 25691, 25703, 25727, 25755 25831, 25839, 25851, 25855, 25883, 25919, 25927, 25959, 25983, 26015, 26047, 26079, 26095, 26111, 26139, 26159 26215, 26239, 26267, 26271, 26279, 26351, 26367, 26395, 26459, 26471, 26495, 26527, 26535, 26559, 26619, 26623 26687, 26695, 26715, 26727, 26735, 26751, 26783, 26791, 26863, 26875, 26879, 26943, 26951, 26983, 27039, 27071 27103, 27119, 27135, 27163, 27183, 27227, 27263, 27291, 27295, 27303, 27327, 27367, 27375, 27387, 27391, 27419 27455, 27463, 27495, 27519, 27559, 27583, 27631, 27647, 27675, 27679, 27739, 27751, 27759, 27775, 27803, 27807 27815, 27839, 27879, 27887, 27899, 27903, 27931, 27967, 27975, 28007, 28063, 28095, 28127, 28143, 28159, 28187 28207, 28287, 28315, 28319, 28391, 28399, 28415, 28443, 28507, 28519, 28575, 28583, 28607, 28671, 28699, 28703 28719, 28743, 28763, 28783, 28831, 28839, 28863, 28911, 28923, 28927, 28975, 28991, 28999, 29031, 29087, 29119 29151, 29167, 29183, 29211, 29231, 29275, 29295, 29311, 29339, 29343, 29351, 29375, 29415, 29423, 29435, 29439 29467, 29511, 29543, 29567, 29631, 29663, 29695, 29723, 29727, 29787, 29799, 29807, 29823, 29855, 29863, 29887 29919, 29927, 29935, 29951, 29979, 29999, 30015, 30023, 30055, 30111, 30143, 30175, 30207, 30235, 30239, 30255 30311, 30335, 30367, 30399, 30439, 30459, 30463, 30491, 30567, 30591, 30631, 30655, 30715, 30719, 30747, 30751 30767, 30811, 30831, 30847, 30879, 30887, 30943, 30959, 30971, 30975, 31003, 31039, 31047, 31079, 31167, 31199 31215, 31231, 31279, 31323, 31335, 31343, 31359, 31387, 31399, 31423, 31463, 31471, 31483, 31487, 31515, 31551 31591, 31615, 31643, 31679, 31711, 31739, 31743, 31771, 31775, 31807, 31847, 31855, 31871, 31899, 31903, 31911 31935, 31983, 31995, 31999, 32027, 32047, 32063, 32103, 32127, 32159, 32191, 32223, 32239, 32255, 32283, 32287 32303, 32347, 32359, 32383, 32415, 32447, 32487, 32495, 32511, 32539, 32603, 32615, 32639, 32703, 32751, 32763 32767 ### Nouveaux résidus couverts en passant de m = 14 à m = 15 Nombre : 159 (parmi les deux enfants modulo 2^15 des résidus non couverts au palier précédent). 191, 231, 935, 1135, 1215, 1351, 1415, 2079, 2239, 2663, 2815, 3067, 3239, 3687, 3707, 3887 3967, 4091, 4123, 4143, 4543, 4711, 4831, 4955, 5087, 5115, 5275, 5595, 5615, 5979, 6271, 6299 6559, 6719, 6975, 6983, 7003, 7015, 7419, 7483, 7919, 8007, 8263, 8447, 8731, 8863, 8871, 9191 9307, 9447, 9755, 9895, 10151, 10175, 10779, 10799, 11375, 11455, 11743, 12319, 12647, 13051, 13083, 13631 13927, 14075, 14207, 14331, 14939, 15259, 15515, 15855, 15963, 16103, 16219, 16687, 16799, 16967, 17007, 17223 17327, 17631, 17711, 17951, 18031, 18111, 18271, 18655, 18687, 18715, 18735, 18855, 18975, 19111, 19215, 19519 19679, 19839, 19995, 20415, 20463, 20543, 20583, 20863, 21039, 21407, 21487, 21567, 21983, 22171, 22431, 22511 22887, 23035, 23167, 23291, 23455, 23871, 23879, 24815, 24923, 25179, 25759, 25767, 26343, 26651, 26671, 26991 27247, 27615, 27695, 27935, 27951, 28191, 28351, 28639, 28799, 28895, 28955, 29503, 29767, 29947, 30079, 30407 30447, 30527, 30783, 30791, 30823, 31391, 31655, 31727, 31815, 31835, 31975, 32295, 32411, 32671, 32679 Annotations (type, horizon, somme A, seuil N) pour les 24 premiers : - r=191 : D ; horizon=8 ; A=14 ; N=1 - r=231 : D ; horizon=7 ; A=14 ; N=1 - r=935 : D ; horizon=7 ; A=14 ; N=1 - r=1135 : D ; horizon=8 ; A=14 ; N=1 - r=1215 : D ; horizon=8 ; A=14 ; N=1 - r=1351 : F ; horizon=9 ; A=14 ; N=5 - r=1415 : D ; horizon=4 ; A=14 ; N=1 - r=2079 : D ; horizon=8 ; A=14 ; N=1 - r=2239 : D ; horizon=7 ; A=14 ; N=1 - r=2663 : D ; horizon=8 ; A=14 ; N=1 - r=2815 : F ; horizon=9 ; A=14 ; N=3 - r=3067 : D ; horizon=8 ; A=14 ; N=2 - r=3239 : F ; horizon=9 ; A=14 ; N=5 - r=3687 : D ; horizon=8 ; A=14 ; N=1 - r=3707 : D ; horizon=5 ; A=14 ; N=1 - r=3887 : D ; horizon=6 ; A=14 ; N=1 - r=3967 : D ; horizon=8 ; A=14 ; N=1 - r=4091 : D ; horizon=8 ; A=14 ; N=2 - r=4123 : F ; horizon=9 ; A=14 ; N=5 - r=4143 : F ; horizon=9 ; A=14 ; N=5 - r=4543 : F ; horizon=9 ; A=14 ; N=3 - r=4711 : D ; horizon=7 ; A=14 ; N=1 - r=4831 : D ; horizon=6 ; A=14 ; N=1 - r=4955 : D ; horizon=8 ; A=14 ; N=2 ### Palier m = 16 (mod 2^16 = 65536) Nombre de résidus impairs non couverts : 2446 sur 32768. 27, 31, 47, 63, 71, 91, 103, 111, 159, 167, 223, 239, 251, 283, 319, 327 359, 415, 447, 479, 495, 511, 559, 603, 639, 667, 671, 703, 743, 751, 763, 767 795, 831, 859, 871, 895, 927, 959, 991, 1007, 1023, 1051, 1055, 1115, 1127, 1151, 1179 1183, 1247, 1255, 1263, 1279, 1307, 1327, 1343, 1383, 1407, 1439, 1471, 1503, 1519, 1535, 1567 1583, 1639, 1647, 1663, 1691, 1695, 1727, 1767, 1791, 1819, 1883, 1887, 1895, 1919, 1951, 1959 2043, 2047, 2111, 2119, 2139, 2151, 2159, 2175, 2207, 2215, 2287, 2299, 2303, 2367, 2375, 2407 2463, 2495, 2527, 2543, 2559, 2651, 2671, 2687, 2715, 2719, 2751, 2791, 2799, 2811, 2843, 2879 2887, 2919, 2943, 3007, 3055, 3071, 3099, 3103, 3135, 3175, 3183, 3199, 3227, 3231, 3263, 3295 3311, 3323, 3327, 3355, 3375, 3391, 3399, 3431, 3455, 3487, 3519, 3567, 3583, 3611, 3615, 3631 3711, 3739, 3775, 3815, 3823, 3839, 3867, 3931, 3943, 3999, 4007, 4031, 4095, 4127, 4167, 4187 4207, 4255, 4263, 4287, 4319, 4335, 4347, 4351, 4379, 4399, 4415, 4423, 4511, 4575, 4591, 4607 4635, 4699, 4719, 4735, 4763, 4767, 4775, 4799, 4839, 4847, 4863, 4891, 4935, 4967, 4991, 5023 5055, 5103, 5119, 5147, 5151, 5183, 5211, 5223, 5247, 5279, 5287, 5343, 5351, 5359, 5375, 5403 5423, 5447, 5479, 5503, 5535, 5567, 5599, 5659, 5663, 5679, 5735, 5759, 5791, 5823, 5863, 5887 5915, 5991, 6015, 6055, 6079, 6139, 6143, 6171, 6175, 6191, 6207, 6235, 6247, 6255, 6303, 6311 6367, 6383, 6395, 6399, 6427, 6463, 6471, 6591, 6623, 6639, 6651, 6655, 6703, 6767, 6811, 6823 6847, 6887, 6895, 6907, 6911, 6939, 7039, 7071, 7103, 7135, 7151, 7163, 7167, 7195, 7231, 7271 7279, 7295, 7323, 7327, 7335, 7359, 7407, 7423, 7451, 7471, 7487, 7527, 7551, 7583, 7615, 7647 7663, 7679, 7707, 7711, 7727, 7783, 7807, 7835, 7839, 7871, 7935, 7963, 8027, 8039, 8063, 8095 8127, 8175, 8191, 8219, 8223, 8239, 8255, 8283, 8295, 8303, 8319, 8351, 8359, 8415, 8443, 8475 8511, 8519, 8539, 8551, 8607, 8639, 8671, 8687, 8703, 8751, 8795, 8831, 8859, 8895, 8935, 8943 8955, 8959, 8987, 9023, 9031, 9051, 9063, 9119, 9151, 9183, 9199, 9215, 9243, 9247, 9287, 9319 9343, 9371, 9375, 9455, 9467, 9471, 9499, 9535, 9543, 9575, 9599, 9631, 9663, 9695, 9711, 9727 9775, 9831, 9855, 9883, 9887, 9967, 9983, 10011, 10087, 10111, 10143, 10235, 10239, 10267, 10287, 10303 10311, 10331, 10343, 10351, 10367, 10399, 10407, 10479, 10491, 10495, 10559, 10567, 10599, 10607, 10687, 10719 10751, 10843, 10863, 10879, 10907, 10911, 10919, 10943, 10983, 10991, 11003, 11007, 11035, 11071, 11079, 11111 11135, 11175, 11199, 11247, 11263, 11291, 11295, 11355, 11367, 11391, 11419, 11423, 11431, 11495, 11503, 11515 11519, 11547, 11583, 11591, 11623, 11679, 11711, 11759, 11775, 11803, 11823, 11903, 11931, 11935, 12007, 12015 12031, 12059, 12123, 12135, 12191, 12199, 12223, 12255, 12287, 12315, 12335, 12359, 12379, 12399, 12415, 12447 12455, 12479, 12511, 12527, 12539, 12571, 12591, 12607, 12615, 12703, 12735, 12767, 12783, 12799, 12827, 12847 12891, 12911, 12927, 12955, 12959, 12967, 12991, 13031, 13039, 13055, 13127, 13159, 13183, 13247, 13279, 13311 13339, 13343, 13383, 13403, 13415, 13423, 13439, 13471, 13479, 13503, 13535, 13543, 13551, 13563, 13567, 13595 13615, 13639, 13671, 13727, 13759, 13791, 13823, 13855, 13871, 13951, 13983, 14015, 14023, 14055, 14063, 14079 14107, 14143, 14183, 14247, 14335, 14363, 14367, 14383, 14399, 14427, 14439, 14447, 14463, 14495, 14503, 14559 14575, 14587, 14591, 14619, 14655, 14663, 14695, 14783, 14815, 14831, 14847, 14895, 14951, 14959, 14975, 15003 15015, 15039, 15079, 15087, 15099, 15103, 15131, 15167, 15207, 15231, 15271, 15295, 15327, 15355, 15359, 15387 15391, 15423, 15431, 15451, 15463, 15471, 15487, 15519, 15527, 15551, 15591, 15599, 15611, 15615, 15643, 15663 15679, 15719, 15743, 15775, 15807, 15871, 15899, 15903, 15911, 15975, 15999, 16027, 16031, 16063, 16111, 16127 16155, 16231, 16255, 16287, 16319, 16367, 16379, 16383, 16411, 16415, 16431, 16447, 16455, 16475, 16487, 16495 16511, 16543, 16551, 16607, 16623, 16635, 16667, 16703, 16711, 16743, 16831, 16863, 16879, 16895, 16943, 16987 17023, 17051, 17055, 17087, 17127, 17135, 17151, 17179, 17215, 17243, 17255, 17279, 17311, 17319, 17343, 17375 17391, 17407, 17435, 17439, 17479, 17499, 17511, 17519, 17535, 17563, 17567, 17599, 17639, 17647, 17659, 17663 17691, 17735, 17791, 17799, 17823, 17855, 17887, 17903, 17919, 17967, 18023, 18047, 18075, 18079, 18151, 18159 18175, 18203, 18267, 18279, 18303, 18335, 18343, 18427, 18431, 18495, 18503, 18523, 18535, 18543, 18559, 18591 18599, 18671, 18683, 18751, 18759, 18791, 18847, 18879, 18911, 18927, 18943, 19047, 19055, 19071, 19099, 19103 19135, 19175, 19183, 19195, 19227, 19263, 19271, 19303, 19327, 19367, 19391, 19439, 19455, 19483, 19487, 19547 19559, 19567, 19583, 19611, 19615, 19623, 19647, 19687, 19695, 19707, 19711, 19739, 19759, 19775, 19783, 19815 19871, 19903, 19951, 19967, 19999, 20015, 20095, 20123, 20127, 20159, 20199, 20207, 20223, 20251, 20315, 20327 20383, 20391, 20475, 20479, 20511, 20551, 20571, 20591, 20639, 20647, 20671, 20703, 20719, 20731, 20735, 20763 20799, 20807, 20895, 20959, 20975, 20991, 21019, 21083, 21095, 21119, 21147, 21151, 21159, 21183, 21215, 21231 21247, 21275, 21319, 21351, 21375, 21439, 21471, 21503, 21531, 21535, 21595, 21607, 21615, 21631, 21663, 21671 21695, 21727, 21735, 21743, 21759, 21787, 21807, 21831, 21863, 21887, 21919, 21951, 22015, 22043, 22047, 22063 22119, 22143, 22175, 22207, 22247, 22271, 22299, 22363, 22375, 22399, 22439, 22463, 22523, 22527, 22555, 22559 22575, 22591, 22619, 22631, 22639, 22683, 22687, 22695, 22767, 22779, 22783, 22847, 22855, 22943, 22975, 23007 23023, 23039, 23087, 23103, 23143, 23151, 23195, 23207, 23271, 23279, 23295, 23323, 23359, 23399, 23423, 23487 23519, 23535, 23547, 23551, 23579, 23583, 23615, 23655, 23663, 23679, 23707, 23711, 23719, 23743, 23791, 23807 23835, 23855, 23911, 23935, 23967, 23999, 24031, 24047, 24063, 24091, 24095, 24111, 24167, 24191, 24219, 24223 24255, 24319, 24347, 24391, 24411, 24423, 24447, 24479, 24511, 24571, 24575, 24603, 24607, 24623, 24647, 24667 24687, 24703, 24735, 24743, 24799, 24827, 24831, 24859, 24895, 24903, 24935, 24991, 25023, 25055, 25071, 25087 25135, 25215, 25243, 25247, 25279, 25319, 25327, 25339, 25343, 25371, 25407, 25415, 25435, 25447, 25471, 25503 25535, 25567, 25583, 25599, 25627, 25631, 25671, 25703, 25727, 25755, 25839, 25851, 25855, 25883, 25919, 25927 25959, 25983, 26015, 26047, 26079, 26095, 26111, 26139, 26159, 26215, 26239, 26271, 26279, 26351, 26367, 26395 26459, 26471, 26495, 26535, 26559, 26619, 26623, 26687, 26695, 26715, 26727, 26735, 26751, 26783, 26791, 26863 26875, 26879, 26943, 26951, 26983, 27039, 27071, 27103, 27119, 27135, 27227, 27263, 27291, 27295, 27303, 27327 27367, 27375, 27387, 27391, 27419, 27455, 27463, 27495, 27519, 27559, 27583, 27631, 27647, 27675, 27679, 27739 27751, 27775, 27803, 27807, 27815, 27879, 27887, 27899, 27903, 27931, 27967, 28007, 28063, 28095, 28127, 28143 28159, 28187, 28207, 28287, 28315, 28319, 28391, 28399, 28415, 28443, 28507, 28519, 28575, 28583, 28607, 28671 28699, 28703, 28719, 28743, 28763, 28783, 28831, 28839, 28863, 28911, 28923, 28927, 28975, 28991, 28999, 29087 29119, 29151, 29167, 29183, 29211, 29231, 29275, 29295, 29311, 29339, 29343, 29351, 29375, 29415, 29423, 29435 29439, 29511, 29543, 29567, 29631, 29663, 29695, 29723, 29727, 29787, 29799, 29807, 29823, 29855, 29887, 29919 29927, 29935, 29951, 29979, 29999, 30015, 30023, 30055, 30111, 30143, 30175, 30207, 30235, 30239, 30255, 30311 30335, 30367, 30399, 30439, 30463, 30491, 30567, 30591, 30631, 30655, 30719, 30747, 30751, 30767, 30811, 30831 30847, 30879, 30887, 30943, 30959, 30971, 30975, 31003, 31039, 31047, 31079, 31167, 31199, 31215, 31231, 31279 31323, 31335, 31343, 31359, 31387, 31399, 31423, 31463, 31471, 31483, 31487, 31515, 31551, 31591, 31615, 31643 31679, 31739, 31743, 31775, 31807, 31847, 31855, 31871, 31899, 31903, 31911, 31935, 31983, 31995, 31999, 32027 32047, 32063, 32103, 32127, 32159, 32191, 32223, 32239, 32255, 32283, 32287, 32303, 32359, 32383, 32415, 32447 32495, 32511, 32539, 32615, 32639, 32703, 32751, 32763, 32767, 32795, 32799, 32815, 32831, 32839, 32859, 32871 32879, 32895, 32927, 32935, 32991, 33007, 33019, 33051, 33071, 33087, 33095, 33127, 33215, 33247, 33263, 33279 33327, 33351, 33371, 33391, 33407, 33435, 33439, 33471, 33511, 33519, 33531, 33535, 33563, 33607, 33627, 33639 33663, 33695, 33711, 33727, 33759, 33775, 33791, 33819, 33823, 33863, 33883, 33895, 33947, 33951, 34023, 34031 34043, 34047, 34075, 34111, 34151, 34175, 34207, 34239, 34271, 34287, 34303, 34351, 34407, 34431, 34459, 34463 34535, 34543, 34559, 34587, 34651, 34663, 34687, 34719, 34727, 34811, 34815, 34879, 34907, 34919, 34927, 34943 34975, 34983, 35039, 35055, 35099, 35119, 35135, 35143, 35175, 35231, 35239, 35263, 35295, 35311, 35327, 35359 35419, 35439, 35455, 35483, 35495, 35519, 35559, 35579, 35599, 35611, 35647, 35655, 35687, 35711, 35751, 35775 35823, 35839, 35867, 35871, 35931, 35943, 35951, 35967, 35995, 35999, 36031, 36071, 36079, 36091, 36095, 36123 36143, 36159, 36167, 36199, 36255, 36287, 36335, 36351, 36383, 36399, 36479, 36507, 36511, 36543, 36583, 36591 36607, 36635, 36699, 36711, 36767, 36847, 36863, 36895, 36927, 36935, 36967, 36975, 37023, 37031, 37087, 37103 37115, 37119, 37147, 37167, 37183, 37191, 37247, 37279, 37343, 37359, 37375, 37403, 37423, 37467, 37487, 37503 37531, 37535, 37543, 37567, 37607, 37615, 37631, 37659, 37703, 37735, 37759, 37823, 37887, 37915, 37919, 37979 37991, 37999, 38015, 38047, 38055, 38079, 38111, 38119, 38127, 38143, 38171, 38191, 38215, 38247, 38271, 38303 38335, 38399, 38427, 38431, 38447, 38503, 38527, 38555, 38559, 38591, 38631, 38655, 38683, 38759, 38783, 38815 38823, 38847, 38895, 38907, 38911, 38939, 38959, 38975, 39003, 39015, 39071, 39079, 39135, 39151, 39163, 39167 39195, 39231, 39239, 39271, 39359, 39391, 39407, 39423, 39471, 39527, 39535, 39551, 39579, 39591, 39615, 39655 39663, 39679, 39707, 39807, 39871, 39903, 39919, 39935, 39963, 39967, 39999, 40039, 40047, 40063, 40091, 40095 40103, 40127, 40175, 40191, 40219, 40239, 40263, 40295, 40319, 40351, 40383, 40415, 40431, 40447, 40475, 40479 40495, 40575, 40603, 40607, 40639, 40703, 40731, 40795, 40807, 40863, 40895, 40943, 40955, 40959, 40987, 40991 41007, 41023, 41051, 41063, 41071, 41087, 41119, 41127, 41183, 41199, 41211, 41243, 41279, 41287, 41319, 41375 41407, 41439, 41455, 41471, 41519, 41599, 41627, 41663, 41703, 41711, 41723, 41727, 41755, 41791, 41799, 41831 41855, 41887, 41919, 41951, 41967, 41983, 42011, 42015, 42055, 42087, 42111, 42151, 42223, 42235, 42239, 42267 42303, 42311, 42343, 42367, 42399, 42431, 42463, 42495, 42543, 42599, 42623, 42651, 42655, 42727, 42735, 42751 42779, 42843, 42855, 42879, 42911, 43003, 43007, 43071, 43079, 43099, 43111, 43119, 43135, 43167, 43175, 43247 43259, 43263, 43327, 43335, 43367, 43423, 43455, 43487, 43503, 43519, 43611, 43647, 43675, 43679, 43687, 43711 43751, 43759, 43771, 43775, 43803, 43839, 43879, 43903, 43943, 43967, 43999, 44015, 44031, 44059, 44063, 44079 44123, 44135, 44159, 44187, 44191, 44199, 44263, 44271, 44283, 44287, 44315, 44319, 44335, 44351, 44359, 44391 44479, 44527, 44543, 44571, 44575, 44591, 44671, 44699, 44703, 44735, 44775, 44783, 44799, 44827, 44891, 44903 44959, 44967, 44991, 45055, 45083, 45103, 45127, 45147, 45167, 45215, 45223, 45247, 45295, 45307, 45311, 45359 45375, 45383, 45471, 45503, 45535, 45551, 45567, 45615, 45659, 45679, 45695, 45723, 45727, 45759, 45799, 45807 45823, 45887, 45895, 45927, 45951, 46015, 46047, 46079, 46107, 46111, 46171, 46183, 46191, 46207, 46239, 46247 46271, 46303, 46311, 46319, 46363, 46383, 46407, 46439, 46463, 46495, 46527, 46559, 46591, 46619, 46623, 46639 46719, 46751, 46783, 46823, 46847, 46875, 46951, 47015, 47039, 47103, 47131, 47135, 47151, 47175, 47195, 47215 47231, 47263, 47271, 47327, 47343, 47355, 47359, 47387, 47423, 47431, 47463, 47551, 47583, 47599, 47615, 47719 47727, 47743, 47771, 47775, 47783, 47807, 47847, 47855, 47867, 47871, 47899, 47935, 47975, 47999, 48095, 48111 48123, 48127, 48155, 48159, 48191, 48231, 48239, 48255, 48287, 48295, 48319, 48367, 48379, 48383, 48411, 48431 48447, 48487, 48511, 48543, 48575, 48607, 48639, 48667, 48671, 48687, 48743, 48767, 48799, 48831, 48879, 48895 48923, 48999, 49023, 49063, 49087, 49135, 49147, 49151, 49179, 49183, 49199, 49215, 49223, 49243, 49255, 49263 49279, 49311, 49319, 49343, 49375, 49383, 49391, 49403, 49435, 49471, 49479, 49599, 49647, 49663, 49711, 49755 49791, 49819, 49823, 49855, 49895, 49903, 49915, 49919, 49947, 49983, 50011, 50023, 50047, 50079, 50111, 50143 50159, 50175, 50203, 50207, 50247, 50267, 50279, 50303, 50331, 50335, 50407, 50415, 50427, 50431, 50459, 50495 50535, 50559, 50591, 50623, 50655, 50671, 50687, 50735, 50791, 50815, 50843, 50847, 50919, 50927, 50943, 50971 51035, 51047, 51071, 51103, 51111, 51195, 51199, 51231, 51263, 51271, 51291, 51303, 51311, 51327, 51359, 51367 51391, 51439, 51451, 51527, 51559, 51615, 51647, 51679, 51695, 51711, 51803, 51823, 51839, 51867, 51871, 51903 51943, 51951, 51963, 51967, 51995, 52031, 52039, 52071, 52095, 52135, 52159, 52207, 52219, 52223, 52251, 52255 52315, 52327, 52335, 52351, 52379, 52383, 52415, 52455, 52463, 52475, 52479, 52507, 52527, 52543, 52551, 52583 52639, 52671, 52719, 52735, 52767, 52783, 52839, 52859, 52863, 52891, 52895, 52927, 52975, 52991, 53019, 53039 53083, 53095, 53119, 53151, 53159, 53247, 53275, 53279, 53295, 53319, 53339, 53359, 53407, 53415, 53439, 53471 53487, 53499, 53503, 53531, 53551, 53567, 53575, 53663, 53695, 53727, 53743, 53759, 53787, 53851, 53871, 53887 53915, 53919, 53927, 53951, 53991, 53999, 54015, 54043, 54087, 54107, 54119, 54143, 54207, 54267, 54271, 54299 54303, 54363, 54375, 54383, 54399, 54427, 54431, 54439, 54463, 54503, 54511, 54527, 54555, 54599, 54631, 54655 54687, 54719, 54747, 54767, 54783, 54811, 54831, 54887, 54911, 54943, 55015, 55039, 55067, 55143, 55167, 55207 55231, 55291, 55295, 55323, 55327, 55343, 55359, 55387, 55399, 55407, 55423, 55455, 55463, 55519, 55535, 55547 55551, 55579, 55615, 55623, 55743, 55775, 55791, 55807, 55855, 55911, 55919, 55963, 55975, 55999, 56039, 56047 56063, 56091, 56135, 56155, 56191, 56255, 56287, 56303, 56315, 56319, 56347, 56351, 56423, 56431, 56447, 56475 56479, 56487, 56511, 56559, 56571, 56575, 56623, 56635, 56679, 56735, 56767, 56799, 56815, 56831, 56859, 56863 56879, 56935, 56959, 56987, 56991, 57023, 57071, 57087, 57115, 57179, 57191, 57215, 57247, 57279, 57327, 57339 57343, 57371, 57375, 57391, 57407, 57435, 57447, 57455, 57471, 57503, 57511, 57567, 57595, 57627, 57663, 57671 57703, 57759, 57791, 57823, 57839, 57855, 57883, 57903, 57983, 58011, 58023, 58047, 58087, 58095, 58107, 58111 58139, 58175, 58183, 58203, 58215, 58239, 58303, 58335, 58343, 58367, 58395, 58399, 58439, 58459, 58495, 58523 58599, 58607, 58619, 58623, 58651, 58687, 58695, 58727, 58751, 58783, 58815, 58847, 58863, 58879, 58927, 58983 59007, 59035, 59039, 59119, 59135, 59163, 59227, 59239, 59263, 59295, 59387, 59391, 59455, 59463, 59483, 59495 59503, 59519, 59551, 59559, 59631, 59643, 59647, 59711, 59719, 59751, 59807, 59839, 59871, 59887, 59903, 59931 59951, 59995, 60031, 60063, 60071, 60095, 60135, 60143, 60155, 60159, 60187, 60223, 60231, 60263, 60287, 60327 60351, 60399, 60415, 60443, 60447, 60507, 60519, 60527, 60543, 60571, 60575, 60583, 60607, 60647, 60655, 60667 60671, 60699, 60735, 60743, 60775, 60831, 60863, 60911, 60927, 60955, 60975, 61055, 61083, 61087, 61159, 61167 61183, 61211, 61275, 61287, 61343, 61351, 61375, 61439, 61467, 61487, 61511, 61531, 61551, 61599, 61607, 61631 61679, 61691, 61695, 61743, 61759, 61799, 61855, 61887, 61919, 61935, 61951, 61979, 61999, 62043, 62063, 62079 62107, 62111, 62119, 62143, 62183, 62191, 62207, 62235, 62279, 62311, 62335, 62399, 62431, 62463, 62491, 62495 62555, 62567, 62575, 62591, 62623, 62631, 62655, 62687, 62695, 62703, 62719, 62747, 62767, 62791, 62823, 62879 62911, 62943, 62975, 63003, 63007, 63023, 63103, 63135, 63167, 63207, 63227, 63231, 63259, 63335, 63399, 63423 63483, 63487, 63519, 63579, 63599, 63615, 63647, 63711, 63727, 63739, 63743, 63771, 63807, 63815, 63847, 63935 63967, 63983, 63999, 64047, 64103, 64111, 64127, 64155, 64167, 64191, 64231, 64239, 64251, 64255, 64283, 64319 64359, 64383, 64447, 64479, 64507, 64511, 64539, 64543, 64575, 64615, 64623, 64639, 64671, 64679, 64703, 64751 64763, 64767, 64795, 64815, 64831, 64871, 64895, 64927, 64959, 64991, 65023, 65051, 65055, 65071, 65115, 65127 65151, 65183, 65215, 65255, 65263, 65279, 65307, 65371, 65383, 65407, 65471, 65519, 65531, 65535 ### Nouveaux résidus couverts en passant de m = 15 à m = 16 Nombre : 244 (parmi les deux enfants modulo 2^16 des résidus non couverts au palier précédent). 127, 303, 583, 623, 839, 943, 1095, 1275, 1775, 2271, 2331, 2351, 2471, 2591, 2727, 2831 2983, 3163, 3303, 3743, 4079, 4159, 4199, 4479, 4655, 5231, 5311, 5631, 5787, 6047, 6127, 6503 6759, 6783, 7199, 7495, 8187, 8431, 9087, 9383, 9959, 10075, 10655, 10735, 11231, 11311, 11551, 11567 11807, 11967, 12543, 13119, 13695, 13851, 14271, 14407, 15007, 15343, 15839, 15919, 16295, 16575, 16615, 17147 17727, 17767, 18463, 18623, 19035, 19199, 19451, 20071, 20091, 20271, 20351, 20507, 20527, 20783, 20927, 21103 21223, 21339, 21499, 21659, 21979, 21999, 22655, 22751, 22811, 23231, 23367, 23387, 23803, 23867, 24303, 24559 24639, 24679, 25115, 25255, 25575, 25691, 25831, 26267, 26527, 27163, 27183, 27759, 27839, 27975, 29031, 29467 29863, 30459, 30715, 31711, 31771, 32347, 32487, 32603, 33183, 33599, 33919, 34015, 34095, 34335, 34415, 34495 34655, 34887, 35067, 35071, 35487, 35567, 35903, 36063, 36223, 36379, 36775, 36799, 36955, 37055, 37791, 37871 37951, 38367, 38943, 39023, 39419, 39675, 39839, 39931, 40255, 40551, 40831, 41307, 41563, 41819, 42139, 42143 42479, 43035, 43055, 43375, 43631, 43847, 44447, 45023, 45183, 45279, 45339, 45595, 45735, 46151, 46331, 46335 46791, 46831, 46911, 47167, 47207, 47663, 48039, 48063, 48199, 48219, 48359, 48679, 48795, 49055, 49511, 49631 50087, 50287, 50367, 50503, 50567, 51519, 51815, 52391, 52967, 53243, 53863, 53983, 54239, 54495, 54575, 54815 54975, 55131, 55451, 55711, 55871, 56127, 56167, 56383, 56603, 56703, 57159, 57415, 57599, 58015, 58271, 58351 58471, 58907, 59047, 59303, 59327, 60059, 60895, 61471, 61767, 62203, 62783, 63079, 63359, 63515, 63535, 63655 64091, 64411, 64667, 65007 Annotations (type, horizon, somme A, seuil N) pour les 24 premiers : - r=127 : D ; horizon=9 ; A=15 ; N=2 - r=303 : D ; horizon=8 ; A=15 ; N=1 - r=583 : D ; horizon=6 ; A=15 ; N=1 - r=623 : D ; horizon=8 ; A=15 ; N=1 - r=839 : D ; horizon=9 ; A=15 ; N=3 - r=943 : D ; horizon=5 ; A=15 ; N=1 - r=1095 : D ; horizon=9 ; A=15 ; N=4 - r=1275 : D ; horizon=9 ; A=15 ; N=3 - r=1775 : D ; horizon=9 ; A=15 ; N=2 - r=2271 : D ; horizon=8 ; A=15 ; N=1 - r=2331 : D ; horizon=8 ; A=15 ; N=1 - r=2351 : D ; horizon=7 ; A=15 ; N=1 - r=2471 : D ; horizon=6 ; A=15 ; N=1 - r=2591 : D ; horizon=7 ; A=15 ; N=1 - r=2727 : D ; horizon=9 ; A=15 ; N=3 - r=2831 : D ; horizon=4 ; A=15 ; N=1 - r=2983 : D ; horizon=9 ; A=15 ; N=3 - r=3163 : D ; horizon=9 ; A=15 ; N=3 - r=3303 : D ; horizon=9 ; A=15 ; N=3 - r=3743 : D ; horizon=9 ; A=15 ; N=2 - r=4079 : D ; horizon=8 ; A=15 ; N=1 - r=4159 : D ; horizon=8 ; A=15 ; N=1 - r=4199 : D ; horizon=8 ; A=15 ; N=1 - r=4479 : D ; horizon=7 ; A=15 ; N=1