This page lists the squares modulo powers of 2, for exponents up to 10. See Modular Arithmetic and Quadratic Residue (table) for background.

There are also lists for the squares modulo N and the powers of 3, 5, 7, 11.

Squares mod 2^1 = 2: 0,1 Squares mod 2^2 = 4: 0,1 Squares mod 2^3 = 8: 0,1,4 Squares mod 2^4 = 16: 0,1,4,9 Squares mod 2^5 = 32: 0,1,4,9,16,17,25 Squares mod 2^6 = 64: 0,1,4,9,16,17,25,33,36,41,49,57 Squares mod 2^7 = 128: 0,1,4,9,16,17,25,33,36,41,49,57,64,65,68,73,81,89,97,100,105,113,121 Squares mod 2^8 = 256: 0,1,4,9,16,17,25,33,36,41,49,57,64,65,68,73,81,89,97,100,105,113,121,129,132,137,144,145,153,161,164,169,177,185,193,196,201,209,217,225,228,233,241,249 Squares mod 2^9 = 512: 0,1,4,9,16,17,25,33,36,41,49,57,64,65,68,73,81,89,97,100,105,113,121,129,132,137,144,145,153,161,164,169,177,185,193,196,201,209,217,225,228,233,241,249,256,257,260,265,272,273,281,289,292,297,305,313,321,324,329,337,345,353,356,361,369,377,385,388,393,400,401,409,417,420,425,433,441,449,452,457,465,473,481,484,489,497,505 Squares mod 2^10 = 1024: 0,1,4,9,16,17,25,33,36,41,49,57,64,65,68,73,81,89,97,100,105,113,121,129,132,137,144,145,153,161,164,169,177,185,193,196,201,209,217,225,228,233,241,249,256,257,260,265,272,273,281,289,292,297,305,313,321,324,329,337,345,353,356,361,369,377,385,388,393,400,401,409,417,420,425,433,441,449,452,457,465,473,481,484,489,497,505,513,516,521,528,529,537,545,548,553,561,569,576,577,580,585,593,601,609,612,617,625,633,641,644,649,656,657,665,673,676,681,689,697,705,708,713,721,729,737,740,745,753,761,769,772,777,784,785,793,801,804,809,817,825,833,836,841,849,857,865,868,873,881,889,897,900,905,912,913,921,929,932,937,945,953,961,964,969,977,985,993,996,1001,1009,1017