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Mar
31
awarded  Necromancer
Jan
3
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 $(c^5+d^5+e^5)$.
Jan
3
revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?
Correction to include missing minus sign.
Dec
31
awarded  Yearling
Dec
22
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).
Dec
22
revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?
Address points in comment by T Piezas.
Dec
22
revised Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?
Clarify relation of answer to known solutions.
Dec
21
answered Congruence properties of $a^5+b^5+c^5+d^5+e^5=0$?
Dec
21
awarded  Constituent
Dec
21
awarded  Caucus
Dec
21
awarded  Necromancer
Oct
1
answered Mathematical breakthroughs
Jun
25
answered Fermat's Last Theorem near misses?
Jun
11
answered What is an example of real application of cubic equations?
Jun
11
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.
Jun
6
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.
Jun
6
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.
Jun
6
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.
May
17
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 $a^3+(-b)^3=c$.
May
16
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 $a^3,b^3 \equiv 1\mod 7$.