Hi all,
\(\displaystyle n=((a^2+b^2)(c^2+d^2)(f^2+2ef-e^2))^2+\)\(\displaystyle \ \ \ \ \ \ ((a^2+b^2)(d^2+2cd-d^2)(e^2+f^2))^2+(b^2+2ab-a^2)(c^2+d^2)(e^2+f^2))^2\)
It is known that
The same is true for the relationship between (c and d) and (e and f)
I have found that \(\displaystyle n\equiv\ 3\ mod\ 4\) by using permutations of the above.
From reasonably extensive computer modelling of n, it also appears that \(\displaystyle n\equiv0\ mod\ 3\) but I can't seem to relate that to the above. Is it obvious why and I'm just suffering brain-freeze?
Thanks for helping thaw me out
\(\displaystyle n=((a^2+b^2)(c^2+d^2)(f^2+2ef-e^2))^2+\)\(\displaystyle \ \ \ \ \ \ ((a^2+b^2)(d^2+2cd-d^2)(e^2+f^2))^2+(b^2+2ab-a^2)(c^2+d^2)(e^2+f^2))^2\)
It is known that
\(\displaystyle a^2+b^2\equiv 1\ mod\ 4\)
\(\displaystyle c^2+d^2\equiv 1\ mod\ 4\)
\(\displaystyle e^2+f^2\equiv 1\ mod\ 4\)
from these we can say that if \(\displaystyle a\equiv(0\ mod\ 4)\ or\ (2\ mod\ 4)\ then\ b\equiv\ (1\ mod\ 4)\ or\ (3\ mod\ 4)\) or vice versa.\(\displaystyle c^2+d^2\equiv 1\ mod\ 4\)
\(\displaystyle e^2+f^2\equiv 1\ mod\ 4\)
The same is true for the relationship between (c and d) and (e and f)
I have found that \(\displaystyle n\equiv\ 3\ mod\ 4\) by using permutations of the above.
From reasonably extensive computer modelling of n, it also appears that \(\displaystyle n\equiv0\ mod\ 3\) but I can't seem to relate that to the above. Is it obvious why and I'm just suffering brain-freeze?
Thanks for helping thaw me out
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