Analysis of Quadratic Residue over Z₁₃₃₁

Quadratic residue (QR)	x such that x2 = QR mod 1331	Occurance
0	0, 121, 242, 363, 484, 605, 726, 847, 968, 1089, 1210	11
1	1, 1330	2
3	578, 753	2
4	2, 1329	2
5	73, 1258	2
9	3, 1328	2
12	175, 1156	2
14	258, 1073	2
15	398, 933	2
16	4, 1327	2
20	146, 1185	2
23	617, 714	2
25	5, 1326	2
26	416, 915	2
27	403, 928	2
31	316, 1015	2
34	285, 1046	2
36	6, 1325	2
37	222, 1109	2
38	37, 1294	2
42	52, 1279	2
45	219, 1112	2
47	225, 1106	2
48	350, 981	2
49	7, 1324	2
53	393, 938	2
56	516, 815	2
58	633, 698	2
59	530, 801	2
60	535, 796	2
64	8, 1323	2
67	155, 1176	2
69	82, 1249	2
70	200, 1131	2
71	216, 1115	2
75	228, 1103	2
78	463, 868	2
80	292, 1039	2
81	9, 1322	2
82	381, 950	2
86	278, 1053	2
89	408, 923	2
91	424, 907	2
92	97, 1234	2
93	301, 1030	2
97	179, 1152	2
100	10, 1321	2
102	314, 1017	2
103	64, 1267	2
104	499, 832	2
108	525, 806	2
111	540, 791	2
113	38, 1293	2
114	90, 1241	2
115	213, 1118	2
119	503, 828	2
121	11, 110, 132, 231, 253, 352, 374, 473, 495, 594, 616, 715, 737, 836, 858, 957, 979, 1078, 1100, 1199, 1221, 1320	22
122	604, 727	2
124	632, 699	2
125	365, 966	2
126	557, 774	2
130	245, 1086	2
133	551, 780	2
135	137, 1194	2
136	570, 761	2
137	480, 851	2
141	388, 943	2
144	12, 1319	2
146	116, 1215	2
147	53, 1278	2
148	444, 887	2
152	74, 1257	2
155	441, 890	2
157	127, 1204	2
158	585, 746	2
159	447, 884	2
163	190, 1141	2
166	507, 824	2
168	104, 1227	2
169	13, 1318	2
170	491, 840	2
174	151, 1180	2
177	210, 1121	2
179	577, 754	2
180	438, 893	2
181	312, 1019	2
185	234, 1097	2
188	450, 881	2
190	39, 1292	2
191	563, 768	2
192	631, 700	2
196	14, 1317	2
199	142, 1189	2
201	413, 918	2
202	354, 977	2
203	466, 865	2
207	520, 811	2
210	197, 1134	2
212	545, 786	2
213	266, 1065	2
214	183, 1148	2
218	421, 910	2
221	615, 716	2
223	435, 896	2
224	299, 1032	2
225	15, 1316	2
229	283, 1048	2
232	65, 1266	2
234	83, 1248	2
235	453, 878	2
236	271, 1060	2
240	261, 1070	2
243	122, 1209	2
245	511, 820	2
246	603, 728	2
247	290, 1041	2
251	487, 844	2
254	54, 1277	2
256	16, 1315	2
257	207, 1124	2
258	367, 964	2
262	630, 701	2
265	593, 738	2
267	237, 1094	2
268	310, 1021	2
269	40, 1291	2
273	168, 1163	2
276	164, 1167	2
278	248, 1083	2
279	383, 948	2
280	400, 931	2
284	432, 899	2
287	98, 1233	2
289	17, 1314	2
290	376, 955	2
291	356, 975	2
295	91, 1240	2
298	395, 936	2
300	456, 875	2
301	75, 1256	2
302	172, 1159	2
306	476, 855	2
309	276, 1055	2
311	160, 1171	2
312	405, 926	2
313	147, 1184	2
317	256, 1075	2
320	584, 747	2
322	534, 797	2
323	614, 717	2
324	18, 1313	2
328	569, 762	2
331	529, 802	2
333	665, 666	2
334	629, 702	2
335	664, 667	2
339	663, 668	2
342	111, 1220	2
344	556, 775	2
345	662, 669	2
346	469, 862	2
350	41, 1290	2
353	661, 670	2
355	204, 1127	2
356	515, 816	2
357	576, 755	2
361	19, 1312	2
363	55, 66, 176, 187, 297, 308, 418, 429, 539, 550, 660, 671, 781, 792, 902, 913, 1023, 1034, 1144, 1155, 1265, 1276	22
364	483, 848	2
366	390, 941	2
367	240, 1091	2
368	194, 1137	2
372	602, 729	2
375	659, 672	2
377	105, 1226	2
378	156, 1175	2
379	117, 1214	2
383	459, 872	2
386	133, 1198	2
388	358, 973	2
389	658, 673	2
390	524, 807	2
394	410, 921	2
397	562, 769	2
399	369, 962	2
400	20, 1311	2
401	84, 1247	2
405	657, 674	2
408	628, 703	2
410	138, 1193	2
411	592, 739	2
412	128, 1203	2
416	333, 998	2
419	331, 1000	2
421	335, 996	2
422	288, 1043	2
423	656, 675	2
427	613, 718	2
430	329, 1002	2
432	281, 1050	2
433	42, 1289	2
434	337, 994	2
438	498, 833	2
441	21, 1310	2
443	655, 676	2
444	251, 1080	2
445	502, 829	2
449	327, 1004	2
452	76, 1255	2
454	544, 787	2
455	339, 992	2
456	180, 1151	2
460	426, 905	2
463	494, 837	2
465	654, 677	2
466	306, 1025	2
467	378, 953	2
471	201, 1130	2
474	56, 1275	2
476	325, 1006	2
477	152, 1179	2
478	92, 1239	2
482	223, 1108	2
484	22, 99, 143, 220, 264, 341, 385, 462, 506, 583, 627, 704, 748, 825, 869, 946, 990, 1067, 1111, 1188, 1232, 1309	22
485	243, 1088	2
487	269, 1062	2
488	123, 1208	2
489	653, 678	2
493	360, 971	2
496	67, 1264	2
498	226, 1105	2
499	519, 812	2
500	601, 730	2
504	217, 1114	2
507	472, 859	2
509	479, 852	2
510	295, 1036	2
511	323, 1008	2
515	652, 679	2
518	43, 1288	2
520	490, 841	2
521	343, 988	2
522	568, 763	2
526	415, 916	2
529	23, 1308	2
531	259, 1072	2
532	229, 1102	2
533	612, 719	2
537	575, 756	2
540	274, 1057	2
542	214, 1117	2
543	651, 680	2
544	191, 1140	2
548	371, 960	2
551	397, 934	2
553	402, 929	2
554	321, 1010	2
555	510, 821	2
559	591, 740	2
562	626, 705	2
564	555, 776	2
565	112, 1219	2
566	345, 986	2
570	85, 1246	2
573	650, 681	2
575	423, 908	2
576	24, 1307	2
577	304, 1027	2
581	184, 1147	2
584	232, 1099	2
586	533, 798	2
587	57, 1274	2
588	106, 1225	2
592	443, 888	2
595	549, 782	2
597	446, 885	2
598	211, 1120	2
599	392, 939	2
603	465, 866	2
605	44, 77, 165, 198, 286, 319, 407, 440, 528, 561, 649, 682, 770, 803, 891, 924, 1012, 1045, 1133, 1166, 1254, 1287	22
606	362, 969	2
608	148, 1183	2
609	486, 845	2
610	169, 1162	2
614	118, 1213	2
617	538, 793	2
619	347, 984	2
620	449, 882	2
621	246, 1085	2
625	25, 1306	2
628	254, 1077	2
630	600, 731	2
631	68, 1263	2
632	161, 1170	2
636	437, 894	2
639	648, 683	2
641	611, 720	2
642	625, 706	2
643	279, 1052	2
647	173, 1158	2
650	582, 749	2
652	380, 951	2
653	134, 1197	2
654	235, 1096	2
658	514, 817	2
661	452, 879	2
663	93, 1238	2
664	317, 1014	2
665	293, 1038	2
669	129, 1202	2
672	208, 1123	2
674	523, 808	2
675	647, 684	2
676	26, 1305	2
680	349, 982	2
683	100, 1231	2
685	434, 897	2
686	475, 856	2
687	139, 1192	2
691	157, 1174	2
694	45, 1286	2
696	302, 1029	2
697	387, 944	2
698	543, 788	2
702	58, 1273	2
705	373, 958	2
707	412, 919	2
708	420, 911	2
709	590, 741	2
713	646, 685	2
716	177, 1154	2
718	567, 764	2
719	574, 757	2
720	455, 876	2
724	624, 707	2
727	364, 967	2
729	27, 1304	2
730	482, 849	2
731	315, 1016	2
735	124, 1207	2
738	188, 1143	2
740	468, 863	2
741	86, 1245	2
742	238, 1093	2
746	267, 1064	2
749	351, 980	2
751	610, 721	2
752	431, 900	2
753	645, 686	2
757	195, 1136	2
760	78, 1253	2
762	599, 732	2
763	262, 1069	2
764	205, 1126	2
768	69, 1262	2
771	144, 1187	2
773	501, 830	2
774	497, 834	2
775	249, 1082	2
779	272, 1059	2
782	153, 1178	2
784	28, 1303	2
785	46, 1285	2
786	554, 777	2
790	113, 1218	2
793	518, 813	2
795	644, 687	2
796	284, 1047	2
797	458, 873	2
801	107, 1224	2
804	505, 826	2
806	313, 1018	2
807	493, 838	2
808	623, 708	2
812	399, 932	2
815	560, 771	2
817	181, 1150	2
818	581, 750	2
819	59, 1272	2
823	300, 1031	2
826	353, 978	2
828	291, 1040	2
829	548, 783	2
830	257, 1074	2
834	404, 927	2
837	428, 903	2
839	643, 688	2
840	394, 937	2
841	29, 1302	2
845	382, 949	2
848	241, 1090	2
850	94, 1237	2
851	119, 1212	2
852	532, 799	2
856	366, 965	2
859	417, 914	2
861	589, 742	2
862	277, 1054	2
863	609, 722	2
867	509, 822	2
870	375, 956	2
872	489, 842	2
873	537, 794	2
874	202, 1129	2
878	47, 1284	2
881	527, 804	2
883	478, 853	2
884	101, 1230	2
885	642, 689	2
889	311, 1020	2
892	461, 870	2
894	622, 709	2
895	471, 860	2
896	598, 733	2
900	30, 1301	2
903	573, 758	2
905	149, 1182	2
906	409, 922	2
907	70, 1261	2
911	355, 976	2
914	87, 1244	2
916	566, 765	2
917	79, 1252	2
918	389, 942	2
922	135, 1196	2
925	221, 1110	2
927	192, 1139	2
928	130, 1201	2
929	224, 1107	2
933	641, 690	2
936	166, 1165	2
938	60, 1271	2
939	218, 1113	2
940	425, 906	2
944	542, 789	2
947	252, 1079	2
949	170, 1161	2
950	185, 1146	2
951	227, 1104	2
955	162, 1169	2
958	298, 1033	2
960	522, 809	2
961	31, 1300	2
962	513, 818	2
966	140, 1191	2
969	485, 846	2
971	215, 1116	2
972	244, 1087	2
973	48, 1283	2
977	608, 723	2
980	309, 1022	2
982	621, 710	2
983	640, 691	2
984	125, 1206	2
988	580, 751	2
991	230, 1101	2
993	368, 963	2
994	174, 1157	2
995	282, 1049	2
999	289, 1042	2
1002	199, 1132	2
1004	357, 974	2
1005	464, 867	2
1006	158, 1173	2
1010	553, 778	2
1013	265, 1066	2
1015	588, 743	2
1016	108, 1223	2
1017	114, 1217	2
1021	212, 1119	2
1024	32, 1299	2
1026	270, 1061	2
1027	559, 772	2
1028	414, 917	2
1032	597, 734	2
1035	639, 692	2
1037	445, 886	2
1038	442, 889	2
1039	95, 1236	2
1043	377, 954	2
1046	384, 947	2
1048	71, 1260	2
1049	233, 1098	2
1050	260, 1071	2
1054	448, 883	2
1057	439, 892	2
1059	61, 1270	2
1060	145, 1186	2
1061	422, 909	2
1065	547, 784	2
1068	474, 857	2
1070	49, 1282	2
1071	178, 1153	2
1072	620, 711	2
1076	80, 1251	2
1079	307, 1024	2
1081	401, 930	2
1082	332, 999	2
1083	334, 997	2
1087	102, 1229	2
1089	33, 88, 154, 209, 275, 330, 396, 451, 517, 572, 638, 693, 759, 814, 880, 935, 1001, 1056, 1122, 1177, 1243, 1298	22
1090	120, 1211	2
1092	336, 995	2
1093	607, 724	2
1094	436, 895	2
1098	481, 850	2
1101	296, 1035	2
1103	500, 831	2
1104	328, 1003	2
1105	359, 972	2
1109	338, 993	2
1112	496, 835	2
1114	247, 1084	2
1115	189, 1142	2
1116	565, 766	2
1120	531, 800	2
1123	406, 925	2
1125	236, 1095	2
1126	504, 827	2
1127	326, 1005	2
1131	536, 795	2
1134	340, 991	2
1136	467, 864	2
1137	255, 1076	2
1138	370, 961	2
1142	454, 877	2
1145	637, 694	2
1147	391, 940	2
1148	196, 1135	2
1149	433, 898	2
1153	492, 839	2
1156	34, 1297	2
1158	324, 1007	2
1159	526, 805	2
1160	579, 752	2
1164	619, 712	2
1167	342, 989	2
1169	50, 1281	2
1170	596, 735	2
1171	587, 744	2
1175	206, 1125	2
1178	287, 1044	2
1180	182, 1149	2
1181	508, 823	2
1182	62, 1269	2
1186	305, 1026	2
1189	131, 1200	2
1191	72, 1259	2
1192	541, 790	2
1193	136, 1195	2
1197	322, 1009	2
1200	419, 912	2
1202	280, 1051	2
1203	636, 695	2
1204	150, 1181	2
1208	344, 987	2
1211	606, 725	2
1213	457, 874	2
1214	361, 970	2
1215	411, 920	2
1219	239, 1092	2
1222	430, 901	2
1224	379, 952	2
1225	35, 1296	2
1226	488, 843	2
1230	96, 1235	2
1233	109, 1222	2
1235	126, 1205	2
1236	552, 779	2
1237	81, 1250	2
1241	558, 773	2
1244	320, 1011	2
1246	115, 1216	2
1247	141, 1190	2
1248	521, 810	2
1252	294, 1037	2
1255	386, 945	2
1257	346, 985	2
1258	618, 713	2
1259	477, 854	2
1263	635, 696	2
1266	89, 1242	2
1268	512, 819	2
1269	167, 1164	2
1270	51, 1280	2
1274	250, 1081	2
1277	571, 760	2
1279	203, 1128	2
1280	163, 1168	2
1281	268, 1063	2
1285	470, 861	2
1288	263, 1068	2
1290	171, 1160	2
1291	372, 959	2
1292	103, 1228	2
1296	36, 1295	2
1299	318, 1013	2
1301	303, 1028	2
1302	460, 871	2
1303	546, 785	2
1307	63, 1268	2
1310	595, 736	2
1312	193, 1138	2
1313	427, 904	2
1314	348, 983	2
1318	564, 767	2
1321	186, 1145	2
1323	159, 1172	2
1324	273, 1058	2
1325	634, 697	2
1329	586, 745	2

Occurance	Quadratic residues	Total QRs with same occurance
2	1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137, 141, 144, 146, 147, 148, 152, 155, 157, 158, 159, 163, 166, 168, 169, 170, 174, 177, 179, 180, 181, 185, 188, 190, 191, 192, 196, 199, 201, 202, 203, 207, 210, 212, 213, 214, 218, 221, 223, 224, 225, 229, 232, 234, 235, 236, 240, 243, 245, 246, 247, 251, 254, 256, 257, 258, 262, 265, 267, 268, 269, 273, 276, 278, 279, 280, 284, 287, 289, 290, 291, 295, 298, 300, 301, 302, 306, 309, 311, 312, 313, 317, 320, 322, 323, 324, 328, 331, 333, 334, 335, 339, 342, 344, 345, 346, 350, 353, 355, 356, 357, 361, 364, 366, 367, 368, 372, 375, 377, 378, 379, 383, 386, 388, 389, 390, 394, 397, 399, 400, 401, 405, 408, 410, 411, 412, 416, 419, 421, 422, 423, 427, 430, 432, 433, 434, 438, 441, 443, 444, 445, 449, 452, 454, 455, 456, 460, 463, 465, 466, 467, 471, 474, 476, 477, 478, 482, 485, 487, 488, 489, 493, 496, 498, 499, 500, 504, 507, 509, 510, 511, 515, 518, 520, 521, 522, 526, 529, 531, 532, 533, 537, 540, 542, 543, 544, 548, 551, 553, 554, 555, 559, 562, 564, 565, 566, 570, 573, 575, 576, 577, 581, 584, 586, 587, 588, 592, 595, 597, 598, 599, 603, 606, 608, 609, 610, 614, 617, 619, 620, 621, 625, 628, 630, 631, 632, 636, 639, 641, 642, 643, 647, 650, 652, 653, 654, 658, 661, 663, 664, 665, 669, 672, 674, 675, 676, 680, 683, 685, 686, 687, 691, 694, 696, 697, 698, 702, 705, 707, 708, 709, 713, 716, 718, 719, 720, 724, 727, 729, 730, 731, 735, 738, 740, 741, 742, 746, 749, 751, 752, 753, 757, 760, 762, 763, 764, 768, 771, 773, 774, 775, 779, 782, 784, 785, 786, 790, 793, 795, 796, 797, 801, 804, 806, 807, 808, 812, 815, 817, 818, 819, 823, 826, 828, 829, 830, 834, 837, 839, 840, 841, 845, 848, 850, 851, 852, 856, 859, 861, 862, 863, 867, 870, 872, 873, 874, 878, 881, 883, 884, 885, 889, 892, 894, 895, 896, 900, 903, 905, 906, 907, 911, 914, 916, 917, 918, 922, 925, 927, 928, 929, 933, 936, 938, 939, 940, 944, 947, 949, 950, 951, 955, 958, 960, 961, 962, 966, 969, 971, 972, 973, 977, 980, 982, 983, 984, 988, 991, 993, 994, 995, 999, 1002, 1004, 1005, 1006, 1010, 1013, 1015, 1016, 1017, 1021, 1024, 1026, 1027, 1028, 1032, 1035, 1037, 1038, 1039, 1043, 1046, 1048, 1049, 1050, 1054, 1057, 1059, 1060, 1061, 1065, 1068, 1070, 1071, 1072, 1076, 1079, 1081, 1082, 1083, 1087, 1090, 1092, 1093, 1094, 1098, 1101, 1103, 1104, 1105, 1109, 1112, 1114, 1115, 1116, 1120, 1123, 1125, 1126, 1127, 1131, 1134, 1136, 1137, 1138, 1142, 1145, 1147, 1148, 1149, 1153, 1156, 1158, 1159, 1160, 1164, 1167, 1169, 1170, 1171, 1175, 1178, 1180, 1181, 1182, 1186, 1189, 1191, 1192, 1193, 1197, 1200, 1202, 1203, 1204, 1208, 1211, 1213, 1214, 1215, 1219, 1222, 1224, 1225, 1226, 1230, 1233, 1235, 1236, 1237, 1241, 1244, 1246, 1247, 1248, 1252, 1255, 1257, 1258, 1259, 1263, 1266, 1268, 1269, 1270, 1274, 1277, 1279, 1280, 1281, 1285, 1288, 1290, 1291, 1292, 1296, 1299, 1301, 1302, 1303, 1307, 1310, 1312, 1313, 1314, 1318, 1321, 1323, 1324, 1325, 1329	605
11	0	1
22	121, 363, 484, 605, 1089	5