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 Mar31 awarded Necromancer Jan3 comment Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? My answer, although accepted, contained an error, now corrected by adding a minus sign before $(c^5+d^5+e^5)$. Jan3 revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? Correction to include missing minus sign. Dec31 awarded Yearling Dec22 comment Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? @TitoPiezasIII I've edited my answer to address your questions (a) and (b). Dec22 revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? Address points in comment by T Piezas. Dec22 revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? Clarify relation of answer to known solutions. Dec21 answered Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$? Dec21 awarded Constituent Dec21 awarded Caucus Dec21 awarded Necromancer Oct1 answered Mathematical breakthroughs Jun25 answered Fermat's Last Theorem near misses? Jun11 answered What is an example of real application of cubic equations? Jun11 comment What is an example of real application of cubic equations? For similar reasons the maximum power that can be generated by a wind turbine is proportional to the cube of the wind speed. Jun6 comment A question about squares It may be significant that the number of remaining rows in 2013 x 2013 equals (2013 - 1)/4 = 503. That may give a clue to a closed form formula for the number of remaining rows in n x n. Jun6 comment A question about squares A simple example is n = 3 (zero cells remaining since 1 is in row 1, 4 in row 2 and 9 in row 3, one quadratic non-residue (2)). n = 2013 is also an example, since there will be many rows not deleted in the lower regions of the square, the intervals between the larger squares being greater than the length of a row. Jun6 comment A question about squares A proposition that is true for any n x n is that if you delete only the columns containing a square, the number of remaining columns will equal the number of quadratic non-residues. If you also delete rows containing a square and count the remaining cells, the number will equal the number of quadratic non-residues only if all rows except one are deleted. This happens to be so for some cases with small numbers, eg 4x4 and 5x5, but is not true in general. May17 comment Positive integer solutions of $a^3 + b^3 = c$ @BrianJ.Fink The modular constraints are best thought of not as flagging possible solutions but as excluding impossible values, either impossible values of $c$ or, for given $c$, impossible values of $a^3,b^3$. The exclusion of values which are impossible for modular reasons from the set of all positive values of $c,a^3,b^3$ can help to find positive solutions faster. It doesn't matter if, as is the case, the modular constraints also apply to $a^3+(-b)^3=c$. May16 comment Positive integer solutions of $a^3 + b^3 = c$ There are also modular constraints which don't exclude values of $c$ but lead to modular conditions on any associated $a^3$ and $b^3$. For example, if $c \equiv 2\mod 7$ then $a^3,b^3 \equiv 1\mod 7$.