Online Factoring Tool
Table of Factors
Table of Primes
Binary values expressed as polynomials can readily be manipulated using the rules of GF(2). The table below shows the irreducible factors of selected GF(2) polynomials of the form xN+1.
Because addition and subtraction are performed identically in GF(2), this is also a table of factors of xN−1.
Note for example that (1001)2 = (1000001) or (x3+1)2 = (x6+1) and in general (xk+1)2 = (x2k+1), such that factors of a term N=k are found duplicated for N=2k.
Online Factoring Tool
Table of Factors
Table of Primes
| N | Irreducible Factors of xN+1 |
| 1 | (11) |
| 2 | (11)2 |
| 3 | (11)(111) |
| 4 | (11)4 |
| 5 | (11)(11111) |
| 6 | (11)2(111)2 |
| 7 | (11)(1011)(1101) |
| 8 | (11)8 |
| 9 | (11)(111)(1001001) |
| 10 | (11)2(11111)2 |
| 11 | (11)(11111111111) |
| 12 | (11)4(111)4 |
| 13 | (11)(1111111111111) |
| 14 | (11)2(1011)2(1101)2 |
| 15 | (11)(111)(10011)(11001)(11111) |
| 16 | (11)16 |
| 17 | (11)(100111001)(111010111) |
| 18 | (11)2(111)2(1001001)2 |
| 19 | (11)(1111111111111111111) |
| 20 | (11)4(11111)4 |
| 21 | (11)(111)(1011)(1101)(1010111)(1110101) |
| 22 | (11)2(11111111111)2 |
| 23 | (11)(101011100011)(110001110101) |
| 24 | (11)8(111)8 |
| 25 | (11)(11111)(100001000010000100001) |
| 26 | (11)2(1111111111111)2 |
| 27 | (11)(111)(1001001)(1000000001000000001) |
| 28 | (11)4(1011)4(1101)4 |
| 29 | (11)(11111111111111111111111111111) |
| 30 | (11)2(111)2(10011)2(11001)2(11111)2 |
| 31 | (11)(100101)(101001)(101111)(110111)(111011)(111101) |
| 33 | (11)(111)(10010101001)(11000100011)(11111111111) |
| 35 | (11)(1011)(1101)(11111)(1011110010111)(1110100111101) |
| 37 | (11)(1111111111111111111111111111111111111) |
| 39 | (11)(111)(1011110001111)(1111000111101)(1111111111111) |
| 41 | (11)(101111100111001111101)(110110100111001011011) |
| 43 | (11)(100111111111001)(101010010010101)(110100010001011) |
| 45 | (11)(111)(10011)(11001)(11111)(1001001)(1000000001001)(1001000000001) |
| 47 | (11)(100011000111011011101111)(111101110110111000110001) |
| 49 | (11)(1011)(1101)(1000000000000010000001)(1000000100000000000001) |
| 51 | (11)(111)(100011011)(100111001)(110011111)(110110001)(111010111)(111110011) |
| 53 | (11)(11111111111111111111111111111111111111111111111111111) |
| 55 | (11)(11111)(11111111111)(101101101001011100111)(111001110100101101101) |
| 57 | (11)(111)(1011100001000011101)(1111011101011101111)(1111111111111111111) |
| 59 | (11)(11111111111111111111111111111111111111111111111111111111111) |
| 61 | (11)(1111111111111111111111111111111111111111111111111111111111111) |
| 63 | (11)(111)(1011)(1101)(1000011)(1001001)(1010111)(1011011)(1100001)(1100111)(1101101)(1110011)(1110101) |
| 65 | (11)(11111)(1000111110001)(1010011100101)(1011101011101)(1101011101011)(1111111111111) |
| 67 | (11)(1111111111111111111111111111111111111111111111111111111111111111111) |
| 69 | (11)(111)(101011100011)(110001110101)(10100110011000001100111)(11100110000011001100101) |
| 71 | (11)(101000011111000000100010000110110011)(110011011000010001000000111110000101) |
| 73 | (11)(1000000011)(1000010111)(1001001011)(1001100101)(1010011001)(1100000001)(1101001001)(1110100001) |
| 75 | (11)(111)(10011)(11001)(11111)(100000000000000100001)(100001000000000000001)(100001000010000100001) |
| 77 | (11)(1011)(1101)(11111111111)(1011100101110010111001110010111)(1110100111001110100111010011101) |
| 79 | (11)(1001100011101111001111010110100000110111)(1110110000010110101111001111011100011001) |
| 81 | (11)(111)(1001001)(1000000001000000001)(1000000000000000000000000001000000000000000000000000001) |
| 83 | (11)(11111111111111111111111111111111111111111111111111111111111111111111111111111111111) |
| 85 | (11)(11111)(100111001)(100111111)(101110111)(101111011)(110001011)(110100011)(110111101)(111010111)(111011101)(111111001) |
| 87 | (11)(111)(10110011110111110001100011111)(11111000110001111101111001101)(11111111111111111111111111111) |
| 89 | (11)(100011000011)(100100110111)(100111101111)(110000110001)(110111111111)(111011001001)(111101111001)(111111111011) |
| 91 | (11)(1011)(1101)(1000101111001)(1001111010001)(1011000000011)(1100000001101)(1101001111011)(1101111001011)(1111111111111) |
| 93 | (11)(111)(100101)(101001)(101111)(110111)(111011)(111101)(10000110101)(10100001011)(10101100001)(11010000101)(11010100111)(11100101011) |
| 95 | (11)(11111)(1111111111111111111)(1000000110100011101111101001011100011)(1100011101001011111011100010110000001) |
| 97 | (11)(1000010110101000100100001000010010001010110100001)(1111100100001101101100011100011011011000010011111) |
| 99 | (11)(111)(1001001)(10010101001)(11000100011)(11111111111)(1000000001000001000001000000001)(1001000000000001000000000001001) |
| 101 | (11)(11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111) |
| 103 | (11)(1011000111000010100111110100000111101111001100001011)(1101000011001111011110000010111110010100001110001101) |
| 105 | (11)(111)(1011)(1101)(10011)(11001)(11111)(1010111)(1110101)(1000101101101)(1001000111011)(1011011010001)(1011110010111)(1101110001001)(1110100111101) |
| 127 | (11)(10000011)(10001001)(10001111)(10010001)(10011101)(10100111)(10101011)(10111001)(10111111)(11000001)(11001011)(11010011)(11010101)(11100101)(11101111)(11110001)(11110111)(11111101) |
| 255 | (11)(111)(10011)(11001)(11111)(100011011)(100011101)(100101011)(100101101)(100111001)(100111111)(101001101)(101011111)(101100011)(101100101)(101101001)(101110001)(101110111)(101111011)(110000111)(110001011)(110001101)(110011111)(110100011)(110101001)(110110001)(110111101)(111000011)(111001111)(111010111)(111011101)(111100111)(111110011)(111110101)(111111001) |
| 511 | (11)(1011)(1101)(1000000011)(1000010001)(1000010111)(1000011011)(1000100001)(1000101101)(1000110011)(1001001011)(1001011001)(1001011111)(1001100101)(1001101001)(1001101111)(1001110111)(1001111101)(1010000111)(1010010101)(1010011001)(1010100011)(1010100101)(1010101111)(1010110111)(1010111101)(1011001111)(1011010001)(1011011011)(1011110101)(1011111001)(1100000001)(1100010011)(1100010101)(1100011111)(1100100011)(1100110001)(1100111011)(1101001001)(1101001111)(1101011011)(1101100001)(1101101011)(1101101101)(1101110011)(1101111111)(1110000101)(1110001111)(1110100001)(1110110101)(1110111001)(1111000111)(1111001011)(1111001101)(1111010101)(1111011001)(1111100011)(1111101001)(1111111011) |
| 1023 | (11)(111)(100101)(101001)(101111)(110111)(111011)(111101)(10000001001)(10000001111)(10000011011)(10000011101)(10000100111)(10000101101)(10000110101)(10001000111)(10001010011)(10001100011)(10001100101)(10001101111)(10010000001)(10010001011)(10010011001)(10010101001)(10010101111)(10011000101)(10011001001)(10011010111)(10011100111)(10011101101)(10011110011)(10011111111)(10100001011)(10100001101)(10100011001)(10100011111)(10100100011)(10100110001)(10100111101)(10101000011)(10101010111)(10101100001)(10101100111)(10101101011)(10110000101)(10110001111)(10110010111)(10110011011)(10110100001)(10110101011)(10110111001)(10111000001)(10111000111)(10111100101)(10111110111)(10111111011)(11000010011)(11000010101)(11000100011)(11000100101)(11000110001)(11000110111)(11001000011)(11001001111)(11001010001)(11001011011)(11001111001)(11001111111)(11010000101)(11010001001)(11010100111)(11010101101)(11010110101)(11010111111)(11011000001)(11011001101)(11011010011)(11011011111)(11011110111)(11011111101)(11100001111)(11100010001)(11100010111)(11100011101)(11100100001)(11100101011)(11100110101)(11100111001)(11101000111)(11101001101)(11101010101)(11101011001)(11101100011)(11101111011)(11101111101)(11110000001)(11110000111)(11110001101)(11110010011)(11110101001)(11110110001)(11111000101)(11111011011)(11111101011)(11111110011)(11111111001)(11111111111) |