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
```