In support of Legendre's conjecture, the following table lists the first hundred square numbers and the prime numbers that are between them. (Of course a single counterexample is enough to disprove).

1	4	2, 3
4	9	5, 7
9	16	11, 13
16	25	17, 19, 23
25	36	29, 31
36	49	37, 41, 43, 47
49	64	53, 59, 61
64	81	67, 71, 73, 79
81	100	83, 89, 97
100	121	101, 103, 107, 109, 113
121	144	127, 131, 137, 139
144	169	149, 151, 157, 163, 167
169	196	173, 179, 181, 191, 193
196	225	197, 199, 211, 223
225	256	227, 229, 233, 239, 241, 251
256	289	257, 263, 269, 271, 277, 281, 283
289	324	293, 307, 311, 313, 317
324	361	331, 337, 347, 349, 353, 359
361	400	367, 373, 379, 383, 389, 397
400	441	401, 409, 419, 421, 431, 433, 439
441	484	443, 449, 457, 461, 463, 467, 479
484	529	487, 491, 499, 503, 509, 521, 523
529	576	541, 547, 557, 563, 569, 571
576	625	577, 587, 593, 599, 601, 607, 613, 617, 619
625	676	631, 641, 643, 647, 653, 659, 661, 673
676	729	677, 683, 691, 701, 709, 719, 727
729	784	733, 739, 743, 751, 757, 761, 769, 773
784	841	787, 797, 809, 811, 821, 823, 827, 829, 839
841	900	853, 857, 859, 863, 877, 881, 883, 887
900	961	907, 911, 919, 929, 937, 941, 947, 953
961	1024	967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021
1024	1089	1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087
1089	1156	1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153
1156	1225	1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223
1225	1296	1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291
1296	1369	1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367
1369	1444	1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439
1444	1521	1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511
1521	1600	1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597
1600	1681	1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669
1681	1764	1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759
1764	1849	1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847
1849	1936	1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933
1936	2025	1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017
2025	2116	2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113
2116	2209	2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207
2209	2304	2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297
2304	2401	2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399
2401	2500	2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477
2500	2601	2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593
2601	2704	2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699
2704	2809	2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803
2809	2916	2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909
2916	3025	2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023
3025	3136	3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121
3136	3249	3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229
3249	3364	3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361
3364	3481	3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469
3481	3600	3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593
3600	3721	3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719
3721	3844	3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833
3844	3969	3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967
3969	4096	3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093
4096	4225	4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219
4225	4356	4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349
4356	4489	4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483
4489	4624	4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621
4624	4761	4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759
4761	4900	4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889
4900	5041	4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039
5041	5184	5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179
5184	5329	5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323
5329	5476	5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471
5476	5625	5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623
5625	5776	5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749
5776	5929	5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927
5929	6084	5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079
6084	6241	6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229
6241	6400	6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397
6400	6561	6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553
6561	6724	6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719
6724	6889	6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883
6889	7056	6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043
7056	7225	7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219
7225	7396	7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393
7396	7569	7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561
7569	7744	7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741
7744	7921	7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919
7921	8100	7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093
8100	8281	8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273
8281	8464	8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461
8464	8649	8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647
8649	8836	8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831
8836	9025	8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013
9025	9216	9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209
9216	9409	9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403
9409	9604	9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601
9604	9801	9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791
9801	10000	9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973
10000	10201	10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193