Adem Relations

This page lists some examples of Adem relations, sorted by degree. The general formula is that when i<2ji \lt 2j, SqiSqj=kZ(jk1i2k)Sqi+jkSqk, \mathrm{Sq}^i \mathrm{Sq}^j = \sum_{k \in \mathbb{Z}} \binom{j - k - 1}{i - 2k} \mathrm{Sq}^{i + j - k} \mathrm{Sq}^k, with the understanding that (nm)=0\binom{n}{m} = 0 if m>nm \gt n or m<0m \lt 0, and Sqk=0\mathrm{Sq}^k = 0 if k<0k \lt 0 (so in practice kk ranges from max(0,ij+1)\max(0, i - j + 1) to i2\lfloor \frac{i}{2}\rfloor).

I learnt the following "derivation rule" from Eric Peterson: all the Adem relations follow from Sq2n1Sqn=0\mathrm{Sq}^{2n - 1} \mathrm{Sq}^n = 0 and asserting that the map SqnSqn1\mathrm{Sq}^{n} \mapsto \mathrm{Sq}^{n - 1} is a derivation.

"i don't know who this version is due to. i learned it from sean tilson who learned it from bob bruner to whom many mysterious things seem obvious" — Eric Peterson

Edited to add: Bob Bruner attributes it to Kristensen's On Secondary Cohomology Operations

A useful interactive calculator can be found at Alex Kruckman's webpage.

Sq1Sq1=0 \mathrm{Sq}^{1}\mathrm{Sq}^{1} = 0 Sq1Sq2=Sq3 \mathrm{Sq}^{1}\mathrm{Sq}^{2} = \mathrm{Sq}^{3} Sq1Sq3=0 \mathrm{Sq}^{1}\mathrm{Sq}^{3} = 0 Sq2Sq2=Sq3Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{2} = \mathrm{Sq}^{3}\mathrm{Sq}^{1} Sq1Sq4=Sq5 \mathrm{Sq}^{1}\mathrm{Sq}^{4} = \mathrm{Sq}^{5} Sq2Sq3=Sq5+Sq4Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{3} = \mathrm{Sq}^{5} + \mathrm{Sq}^{4}\mathrm{Sq}^{1} Sq3Sq2=0 \mathrm{Sq}^{3}\mathrm{Sq}^{2} = 0 Sq1Sq5=0 \mathrm{Sq}^{1}\mathrm{Sq}^{5} = 0 Sq2Sq4=Sq6+Sq5Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{4} = \mathrm{Sq}^{6} + \mathrm{Sq}^{5}\mathrm{Sq}^{1} Sq3Sq3=Sq5Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{3} = \mathrm{Sq}^{5}\mathrm{Sq}^{1} Sq1Sq6=Sq7 \mathrm{Sq}^{1}\mathrm{Sq}^{6} = \mathrm{Sq}^{7} Sq2Sq5=Sq6Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{5} = \mathrm{Sq}^{6}\mathrm{Sq}^{1} Sq3Sq4=Sq7 \mathrm{Sq}^{3}\mathrm{Sq}^{4} = \mathrm{Sq}^{7} Sq4Sq3=Sq5Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{3} = \mathrm{Sq}^{5}\mathrm{Sq}^{2} Sq1Sq7=0 \mathrm{Sq}^{1}\mathrm{Sq}^{7} = 0 Sq2Sq6=Sq7Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{6} = \mathrm{Sq}^{7}\mathrm{Sq}^{1} Sq3Sq5=Sq7Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{5} = \mathrm{Sq}^{7}\mathrm{Sq}^{1} Sq4Sq4=Sq7Sq1+Sq6Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{1} + \mathrm{Sq}^{6}\mathrm{Sq}^{2} Sq5Sq3=0 \mathrm{Sq}^{5}\mathrm{Sq}^{3} = 0 Sq1Sq8=Sq9 \mathrm{Sq}^{1}\mathrm{Sq}^{8} = \mathrm{Sq}^{9} Sq2Sq7=Sq9+Sq8Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{7} = \mathrm{Sq}^{9} + \mathrm{Sq}^{8}\mathrm{Sq}^{1} Sq3Sq6=0 \mathrm{Sq}^{3}\mathrm{Sq}^{6} = 0 Sq4Sq5=Sq9+Sq8Sq1+Sq7Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{5} = \mathrm{Sq}^{9} + \mathrm{Sq}^{8}\mathrm{Sq}^{1} + \mathrm{Sq}^{7}\mathrm{Sq}^{2} Sq5Sq4=Sq7Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{2} Sq1Sq9=0 \mathrm{Sq}^{1}\mathrm{Sq}^{9} = 0 Sq2Sq8=Sq10+Sq9Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{8} = \mathrm{Sq}^{10} + \mathrm{Sq}^{9}\mathrm{Sq}^{1} Sq3Sq7=Sq9Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{7} = \mathrm{Sq}^{9}\mathrm{Sq}^{1} Sq4Sq6=Sq10+Sq8Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{6} = \mathrm{Sq}^{10} + \mathrm{Sq}^{8}\mathrm{Sq}^{2} Sq5Sq5=Sq9Sq1 \mathrm{Sq}^{5}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{1} Sq6Sq4=Sq7Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{4} = \mathrm{Sq}^{7}\mathrm{Sq}^{3} Sq1Sq10=Sq11 \mathrm{Sq}^{1}\mathrm{Sq}^{10} = \mathrm{Sq}^{11} Sq2Sq9=Sq10Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{9} = \mathrm{Sq}^{10}\mathrm{Sq}^{1} Sq3Sq8=Sq11 \mathrm{Sq}^{3}\mathrm{Sq}^{8} = \mathrm{Sq}^{11} Sq4Sq7=Sq11+Sq9Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{7} = \mathrm{Sq}^{11} + \mathrm{Sq}^{9}\mathrm{Sq}^{2} Sq5Sq6=Sq11+Sq9Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{6} = \mathrm{Sq}^{11} + \mathrm{Sq}^{9}\mathrm{Sq}^{2} Sq6Sq5=Sq9Sq2+Sq8Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{2} + \mathrm{Sq}^{8}\mathrm{Sq}^{3} Sq7Sq4=0 \mathrm{Sq}^{7}\mathrm{Sq}^{4} = 0 Sq1Sq11=0 \mathrm{Sq}^{1}\mathrm{Sq}^{11} = 0 Sq2Sq10=Sq11Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{10} = \mathrm{Sq}^{11}\mathrm{Sq}^{1} Sq3Sq9=Sq11Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{9} = \mathrm{Sq}^{11}\mathrm{Sq}^{1} Sq4Sq8=Sq12+Sq11Sq1+Sq10Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{8} = \mathrm{Sq}^{12} + \mathrm{Sq}^{11}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{2} Sq5Sq7=0 \mathrm{Sq}^{5}\mathrm{Sq}^{7} = 0 Sq6Sq6=Sq11Sq1+Sq10Sq2+Sq9Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{2} + \mathrm{Sq}^{9}\mathrm{Sq}^{3} Sq7Sq5=Sq9Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{3} Sq1Sq12=Sq13 \mathrm{Sq}^{1}\mathrm{Sq}^{12} = \mathrm{Sq}^{13} Sq2Sq11=Sq13+Sq12Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{11} = \mathrm{Sq}^{13} + \mathrm{Sq}^{12}\mathrm{Sq}^{1} Sq3Sq10=0 \mathrm{Sq}^{3}\mathrm{Sq}^{10} = 0 Sq4Sq9=Sq12Sq1+Sq11Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{9} = \mathrm{Sq}^{12}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{2} Sq5Sq8=Sq13+Sq11Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{8} = \mathrm{Sq}^{13} + \mathrm{Sq}^{11}\mathrm{Sq}^{2} Sq6Sq7=Sq13+Sq12Sq1+Sq10Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{7} = \mathrm{Sq}^{13} + \mathrm{Sq}^{12}\mathrm{Sq}^{1} + \mathrm{Sq}^{10}\mathrm{Sq}^{3} Sq7Sq6=Sq11Sq2 \mathrm{Sq}^{7}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{2} Sq8Sq5=Sq9Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{5} = \mathrm{Sq}^{9}\mathrm{Sq}^{4} Sq1Sq13=0 \mathrm{Sq}^{1}\mathrm{Sq}^{13} = 0 Sq2Sq12=Sq14+Sq13Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{12} = \mathrm{Sq}^{14} + \mathrm{Sq}^{13}\mathrm{Sq}^{1} Sq3Sq11=Sq13Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{11} = \mathrm{Sq}^{13}\mathrm{Sq}^{1} Sq4Sq10=Sq12Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{10} = \mathrm{Sq}^{12}\mathrm{Sq}^{2} Sq5Sq9=Sq13Sq1 \mathrm{Sq}^{5}\mathrm{Sq}^{9} = \mathrm{Sq}^{13}\mathrm{Sq}^{1} Sq6Sq8=Sq14+Sq13Sq1+Sq11Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{8} = \mathrm{Sq}^{14} + \mathrm{Sq}^{13}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{3} Sq7Sq7=Sq13Sq1+Sq11Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{1} + \mathrm{Sq}^{11}\mathrm{Sq}^{3} Sq8Sq6=Sq11Sq3+Sq10Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{3} + \mathrm{Sq}^{10}\mathrm{Sq}^{4} Sq9Sq5=0 \mathrm{Sq}^{9}\mathrm{Sq}^{5} = 0 Sq1Sq14=Sq15 \mathrm{Sq}^{1}\mathrm{Sq}^{14} = \mathrm{Sq}^{15} Sq2Sq13=Sq14Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{13} = \mathrm{Sq}^{14}\mathrm{Sq}^{1} Sq3Sq12=Sq15 \mathrm{Sq}^{3}\mathrm{Sq}^{12} = \mathrm{Sq}^{15} Sq4Sq11=Sq13Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{11} = \mathrm{Sq}^{13}\mathrm{Sq}^{2} Sq5Sq10=Sq13Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{10} = \mathrm{Sq}^{13}\mathrm{Sq}^{2} Sq6Sq9=Sq14Sq1+Sq13Sq2+Sq12Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{9} = \mathrm{Sq}^{14}\mathrm{Sq}^{1} + \mathrm{Sq}^{13}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{3} Sq7Sq8=Sq15 \mathrm{Sq}^{7}\mathrm{Sq}^{8} = \mathrm{Sq}^{15} Sq8Sq7=Sq13Sq2+Sq12Sq3+Sq11Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{3} + \mathrm{Sq}^{11}\mathrm{Sq}^{4} Sq9Sq6=Sq11Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{4} Sq1Sq15=0 \mathrm{Sq}^{1}\mathrm{Sq}^{15} = 0 Sq2Sq14=Sq15Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{14} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} Sq3Sq13=Sq15Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{13} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} Sq4Sq12=Sq15Sq1+Sq14Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{14}\mathrm{Sq}^{2} Sq5Sq11=0 \mathrm{Sq}^{5}\mathrm{Sq}^{11} = 0 Sq6Sq10=Sq14Sq2+Sq13Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{10} = \mathrm{Sq}^{14}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{3} Sq7Sq9=Sq15Sq1+Sq13Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{9} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{13}\mathrm{Sq}^{3} Sq8Sq8=Sq15Sq1+Sq14Sq2+Sq12Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{1} + \mathrm{Sq}^{14}\mathrm{Sq}^{2} + \mathrm{Sq}^{12}\mathrm{Sq}^{4} Sq9Sq7=Sq13Sq3 \mathrm{Sq}^{9}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{3} Sq10Sq6=Sq11Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{6} = \mathrm{Sq}^{11}\mathrm{Sq}^{5} Sq1Sq16=Sq17 \mathrm{Sq}^{1}\mathrm{Sq}^{16} = \mathrm{Sq}^{17} Sq2Sq15=Sq17+Sq16Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{15} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1} Sq3Sq14=0 \mathrm{Sq}^{3}\mathrm{Sq}^{14} = 0 Sq4Sq13=Sq17+Sq16Sq1+Sq15Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{13} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1} + \mathrm{Sq}^{15}\mathrm{Sq}^{2} Sq5Sq12=Sq15Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{2} Sq6Sq11=Sq14Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{11} = \mathrm{Sq}^{14}\mathrm{Sq}^{3} Sq7Sq10=Sq15Sq2 \mathrm{Sq}^{7}\mathrm{Sq}^{10} = \mathrm{Sq}^{15}\mathrm{Sq}^{2} Sq8Sq9=Sq17+Sq16Sq1+Sq15Sq2+Sq13Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{9} = \mathrm{Sq}^{17} + \mathrm{Sq}^{16}\mathrm{Sq}^{1} + \mathrm{Sq}^{15}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{4} Sq9Sq8=Sq15Sq2+Sq13Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{2} + \mathrm{Sq}^{13}\mathrm{Sq}^{4} Sq10Sq7=Sq13Sq4+Sq12Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{4} + \mathrm{Sq}^{12}\mathrm{Sq}^{5} Sq11Sq6=0 \mathrm{Sq}^{11}\mathrm{Sq}^{6} = 0 Sq1Sq17=0 \mathrm{Sq}^{1}\mathrm{Sq}^{17} = 0 Sq2Sq16=Sq18+Sq17Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{16} = \mathrm{Sq}^{18} + \mathrm{Sq}^{17}\mathrm{Sq}^{1} Sq3Sq15=Sq17Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{15} = \mathrm{Sq}^{17}\mathrm{Sq}^{1} Sq4Sq14=Sq18+Sq16Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{14} = \mathrm{Sq}^{18} + \mathrm{Sq}^{16}\mathrm{Sq}^{2} Sq5Sq13=Sq17Sq1 \mathrm{Sq}^{5}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{1} Sq6Sq12=Sq15Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{12} = \mathrm{Sq}^{15}\mathrm{Sq}^{3} Sq7Sq11=Sq15Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{11} = \mathrm{Sq}^{15}\mathrm{Sq}^{3} Sq8Sq10=Sq18+Sq16Sq2+Sq15Sq3+Sq14Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{10} = \mathrm{Sq}^{18} + \mathrm{Sq}^{16}\mathrm{Sq}^{2} + \mathrm{Sq}^{15}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{4} Sq9Sq9=Sq17Sq1 \mathrm{Sq}^{9}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{1} Sq10Sq8=Sq15Sq3+Sq14Sq4+Sq13Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{4} + \mathrm{Sq}^{13}\mathrm{Sq}^{5} Sq11Sq7=Sq13Sq5 \mathrm{Sq}^{11}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{5} Sq1Sq18=Sq19 \mathrm{Sq}^{1}\mathrm{Sq}^{18} = \mathrm{Sq}^{19} Sq2Sq17=Sq18Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{17} = \mathrm{Sq}^{18}\mathrm{Sq}^{1} Sq3Sq16=Sq19 \mathrm{Sq}^{3}\mathrm{Sq}^{16} = \mathrm{Sq}^{19} Sq4Sq15=Sq19+Sq17Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{15} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2} Sq5Sq14=Sq19+Sq17Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{14} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2} Sq6Sq13=Sq17Sq2+Sq16Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{16}\mathrm{Sq}^{3} Sq7Sq12=0 \mathrm{Sq}^{7}\mathrm{Sq}^{12} = 0 Sq8Sq11=Sq19+Sq16Sq3+Sq15Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{11} = \mathrm{Sq}^{19} + \mathrm{Sq}^{16}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{4} Sq9Sq10=Sq19+Sq17Sq2+Sq15Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{10} = \mathrm{Sq}^{19} + \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{15}\mathrm{Sq}^{4} Sq10Sq9=Sq17Sq2+Sq16Sq3+Sq14Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{2} + \mathrm{Sq}^{16}\mathrm{Sq}^{3} + \mathrm{Sq}^{14}\mathrm{Sq}^{5} Sq11Sq8=Sq15Sq4 \mathrm{Sq}^{11}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{4} Sq12Sq7=Sq13Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{7} = \mathrm{Sq}^{13}\mathrm{Sq}^{6} Sq1Sq19=0 \mathrm{Sq}^{1}\mathrm{Sq}^{19} = 0 Sq2Sq18=Sq19Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{18} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} Sq3Sq17=Sq19Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{17} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} Sq4Sq16=Sq20+Sq19Sq1+Sq18Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{16} = \mathrm{Sq}^{20} + \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2} Sq5Sq15=0 \mathrm{Sq}^{5}\mathrm{Sq}^{15} = 0 Sq6Sq14=Sq19Sq1+Sq18Sq2+Sq17Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{14} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2} + \mathrm{Sq}^{17}\mathrm{Sq}^{3} Sq7Sq13=Sq17Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{13} = \mathrm{Sq}^{17}\mathrm{Sq}^{3} Sq8Sq12=Sq20+Sq16Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{12} = \mathrm{Sq}^{20} + \mathrm{Sq}^{16}\mathrm{Sq}^{4} Sq9Sq11=Sq17Sq3 \mathrm{Sq}^{9}\mathrm{Sq}^{11} = \mathrm{Sq}^{17}\mathrm{Sq}^{3} Sq10Sq10=Sq19Sq1+Sq18Sq2+Sq17Sq3+Sq15Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{2} + \mathrm{Sq}^{17}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{5} Sq11Sq9=Sq17Sq3+Sq15Sq5 \mathrm{Sq}^{11}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{3} + \mathrm{Sq}^{15}\mathrm{Sq}^{5} Sq12Sq8=Sq15Sq5+Sq14Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{5} + \mathrm{Sq}^{14}\mathrm{Sq}^{6} Sq13Sq7=0 \mathrm{Sq}^{13}\mathrm{Sq}^{7} = 0 Sq1Sq20=Sq21 \mathrm{Sq}^{1}\mathrm{Sq}^{20} = \mathrm{Sq}^{21} Sq2Sq19=Sq21+Sq20Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{19} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1} Sq3Sq18=0 \mathrm{Sq}^{3}\mathrm{Sq}^{18} = 0 Sq4Sq17=Sq20Sq1+Sq19Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{17} = \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{2} Sq5Sq16=Sq21+Sq19Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{16} = \mathrm{Sq}^{21} + \mathrm{Sq}^{19}\mathrm{Sq}^{2} Sq6Sq15=Sq21+Sq20Sq1+Sq18Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{15} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{3} Sq7Sq14=Sq19Sq2 \mathrm{Sq}^{7}\mathrm{Sq}^{14} = \mathrm{Sq}^{19}\mathrm{Sq}^{2} Sq8Sq13=Sq21+Sq17Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{13} = \mathrm{Sq}^{21} + \mathrm{Sq}^{17}\mathrm{Sq}^{4} Sq9Sq12=Sq21+Sq17Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{12} = \mathrm{Sq}^{21} + \mathrm{Sq}^{17}\mathrm{Sq}^{4} Sq10Sq11=Sq21+Sq20Sq1+Sq18Sq3+Sq17Sq4+Sq16Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{11} = \mathrm{Sq}^{21} + \mathrm{Sq}^{20}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{3} + \mathrm{Sq}^{17}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{5} Sq11Sq10=Sq19Sq2 \mathrm{Sq}^{11}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{2} Sq12Sq9=Sq17Sq4+Sq16Sq5+Sq15Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{5} + \mathrm{Sq}^{15}\mathrm{Sq}^{6} Sq13Sq8=Sq15Sq6 \mathrm{Sq}^{13}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{6} Sq1Sq21=0 \mathrm{Sq}^{1}\mathrm{Sq}^{21} = 0 Sq2Sq20=Sq22+Sq21Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{20} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1} Sq3Sq19=Sq21Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{19} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} Sq4Sq18=Sq20Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{18} = \mathrm{Sq}^{20}\mathrm{Sq}^{2} Sq5Sq17=Sq21Sq1 \mathrm{Sq}^{5}\mathrm{Sq}^{17} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} Sq6Sq16=Sq22+Sq21Sq1+Sq19Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{16} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} Sq7Sq15=Sq21Sq1+Sq19Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{15} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} Sq8Sq14=Sq22+Sq19Sq3+Sq18Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{14} = \mathrm{Sq}^{22} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{18}\mathrm{Sq}^{4} Sq9Sq13=0 \mathrm{Sq}^{9}\mathrm{Sq}^{13} = 0 Sq10Sq12=Sq22+Sq21Sq1+Sq18Sq4+Sq17Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{12} = \mathrm{Sq}^{22} + \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{5} Sq11Sq11=Sq21Sq1+Sq19Sq3+Sq17Sq5 \mathrm{Sq}^{11}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{17}\mathrm{Sq}^{5} Sq12Sq10=Sq19Sq3+Sq18Sq4+Sq16Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{3} + \mathrm{Sq}^{18}\mathrm{Sq}^{4} + \mathrm{Sq}^{16}\mathrm{Sq}^{6} Sq13Sq9=Sq17Sq5 \mathrm{Sq}^{13}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{5} Sq14Sq8=Sq15Sq7 \mathrm{Sq}^{14}\mathrm{Sq}^{8} = \mathrm{Sq}^{15}\mathrm{Sq}^{7} Sq1Sq22=Sq23 \mathrm{Sq}^{1}\mathrm{Sq}^{22} = \mathrm{Sq}^{23} Sq2Sq21=Sq22Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{21} = \mathrm{Sq}^{22}\mathrm{Sq}^{1} Sq3Sq20=Sq23 \mathrm{Sq}^{3}\mathrm{Sq}^{20} = \mathrm{Sq}^{23} Sq4Sq19=Sq21Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{19} = \mathrm{Sq}^{21}\mathrm{Sq}^{2} Sq5Sq18=Sq21Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{18} = \mathrm{Sq}^{21}\mathrm{Sq}^{2} Sq6Sq17=Sq22Sq1+Sq21Sq2+Sq20Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{17} = \mathrm{Sq}^{22}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3} Sq7Sq16=Sq23 \mathrm{Sq}^{7}\mathrm{Sq}^{16} = \mathrm{Sq}^{23} Sq8Sq15=Sq23+Sq21Sq2+Sq20Sq3+Sq19Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{15} = \mathrm{Sq}^{23} + \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3} + \mathrm{Sq}^{19}\mathrm{Sq}^{4} Sq9Sq14=Sq23+Sq19Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{14} = \mathrm{Sq}^{23} + \mathrm{Sq}^{19}\mathrm{Sq}^{4} Sq10Sq13=Sq22Sq1+Sq18Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{13} = \mathrm{Sq}^{22}\mathrm{Sq}^{1} + \mathrm{Sq}^{18}\mathrm{Sq}^{5} Sq11Sq12=Sq23+Sq19Sq4 \mathrm{Sq}^{11}\mathrm{Sq}^{12} = \mathrm{Sq}^{23} + \mathrm{Sq}^{19}\mathrm{Sq}^{4} Sq12Sq11=Sq21Sq2+Sq20Sq3+Sq19Sq4+Sq17Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{3} + \mathrm{Sq}^{19}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{6} Sq13Sq10=Sq19Sq4+Sq17Sq6 \mathrm{Sq}^{13}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{4} + \mathrm{Sq}^{17}\mathrm{Sq}^{6} Sq14Sq9=Sq17Sq6+Sq16Sq7 \mathrm{Sq}^{14}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{6} + \mathrm{Sq}^{16}\mathrm{Sq}^{7} Sq15Sq8=0 \mathrm{Sq}^{15}\mathrm{Sq}^{8} = 0 Sq1Sq23=0 \mathrm{Sq}^{1}\mathrm{Sq}^{23} = 0 Sq2Sq22=Sq23Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{22} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} Sq3Sq21=Sq23Sq1 \mathrm{Sq}^{3}\mathrm{Sq}^{21} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} Sq4Sq20=Sq23Sq1+Sq22Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{20} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2} Sq5Sq19=0 \mathrm{Sq}^{5}\mathrm{Sq}^{19} = 0 Sq6Sq18=Sq22Sq2+Sq21Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{18} = \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{3} Sq7Sq17=Sq23Sq1+Sq21Sq3 \mathrm{Sq}^{7}\mathrm{Sq}^{17} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{3} Sq8Sq16=Sq24+Sq23Sq1+Sq22Sq2+Sq20Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{16} = \mathrm{Sq}^{24} + \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{4} Sq9Sq15=Sq21Sq3 \mathrm{Sq}^{9}\mathrm{Sq}^{15} = \mathrm{Sq}^{21}\mathrm{Sq}^{3} Sq10Sq14=Sq23Sq1+Sq19Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{14} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{5} Sq11Sq13=Sq23Sq1+Sq19Sq5 \mathrm{Sq}^{11}\mathrm{Sq}^{13} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{19}\mathrm{Sq}^{5} Sq12Sq12=Sq23Sq1+Sq22Sq2+Sq20Sq4+Sq19Sq5+Sq18Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{12} = \mathrm{Sq}^{23}\mathrm{Sq}^{1} + \mathrm{Sq}^{22}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{4} + \mathrm{Sq}^{19}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{6} Sq13Sq11=Sq21Sq3 \mathrm{Sq}^{13}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{3} Sq14Sq10=Sq19Sq5+Sq18Sq6+Sq17Sq7 \mathrm{Sq}^{14}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{6} + \mathrm{Sq}^{17}\mathrm{Sq}^{7} Sq15Sq9=Sq17Sq7 \mathrm{Sq}^{15}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{7} Sq1Sq24=Sq25 \mathrm{Sq}^{1}\mathrm{Sq}^{24} = \mathrm{Sq}^{25} Sq2Sq23=Sq25+Sq24Sq1 \mathrm{Sq}^{2}\mathrm{Sq}^{23} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} Sq3Sq22=0 \mathrm{Sq}^{3}\mathrm{Sq}^{22} = 0 Sq4Sq21=Sq25+Sq24Sq1+Sq23Sq2 \mathrm{Sq}^{4}\mathrm{Sq}^{21} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} Sq5Sq20=Sq23Sq2 \mathrm{Sq}^{5}\mathrm{Sq}^{20} = \mathrm{Sq}^{23}\mathrm{Sq}^{2} Sq6Sq19=Sq22Sq3 \mathrm{Sq}^{6}\mathrm{Sq}^{19} = \mathrm{Sq}^{22}\mathrm{Sq}^{3} Sq7Sq18=Sq23Sq2 \mathrm{Sq}^{7}\mathrm{Sq}^{18} = \mathrm{Sq}^{23}\mathrm{Sq}^{2} Sq8Sq17=Sq24Sq1+Sq23Sq2+Sq21Sq4 \mathrm{Sq}^{8}\mathrm{Sq}^{17} = \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} Sq9Sq16=Sq25+Sq23Sq2+Sq21Sq4 \mathrm{Sq}^{9}\mathrm{Sq}^{16} = \mathrm{Sq}^{25} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} Sq10Sq15=Sq25+Sq24Sq1+Sq21Sq4+Sq20Sq5 \mathrm{Sq}^{10}\mathrm{Sq}^{15} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{20}\mathrm{Sq}^{5} Sq11Sq14=0 \mathrm{Sq}^{11}\mathrm{Sq}^{14} = 0 Sq12Sq13=Sq25+Sq24Sq1+Sq23Sq2+Sq20Sq5+Sq19Sq6 \mathrm{Sq}^{12}\mathrm{Sq}^{13} = \mathrm{Sq}^{25} + \mathrm{Sq}^{24}\mathrm{Sq}^{1} + \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{20}\mathrm{Sq}^{5} + \mathrm{Sq}^{19}\mathrm{Sq}^{6} Sq13Sq12=Sq23Sq2+Sq21Sq4+Sq19Sq6 \mathrm{Sq}^{13}\mathrm{Sq}^{12} = \mathrm{Sq}^{23}\mathrm{Sq}^{2} + \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{19}\mathrm{Sq}^{6} Sq14Sq11=Sq21Sq4+Sq20Sq5+Sq18Sq7 \mathrm{Sq}^{14}\mathrm{Sq}^{11} = \mathrm{Sq}^{21}\mathrm{Sq}^{4} + \mathrm{Sq}^{20}\mathrm{Sq}^{5} + \mathrm{Sq}^{18}\mathrm{Sq}^{7} Sq15Sq10=Sq19Sq6 \mathrm{Sq}^{15}\mathrm{Sq}^{10} = \mathrm{Sq}^{19}\mathrm{Sq}^{6} Sq16Sq9=Sq17Sq8 \mathrm{Sq}^{16}\mathrm{Sq}^{9} = \mathrm{Sq}^{17}\mathrm{Sq}^{8}