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2h

comment 
Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$
Seva: It doesn't seem to me that $p$ is assumed to be $1\pmod 8$. When $q$ divides $a$ (which is the example you raised) then the biquadratic symbol is determined by $(\frac{2}{q})$. I'm not an exper, and it may be safest to consult Lemmermeyer's book. 
3h

comment 
Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$
The sign doesn't matter. If you multiply $(\frac{\sigma(b+\sigma)}{q})$ and $(\frac{\sigma(b\sigma)}{q})$ together, you get $(\frac{\sigma^2(b\sigma^2)}{q}) = (\frac{a^2}{q})=1$. 
4h

comment 
Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$
An answer is given in the wikipedia page you linked in the section under Dirichlet. If $p\equiv \sigma^2 \pmod q$ then $(\frac{q}{p})_4=(\frac{\sigma(b+\sigma)}{q})$. A reference is given to Lemmermeyer's book. 
16h

reviewed  Leave Closed Calculating the quotient group $\mathbb{Z}\times\mathbb{Z}/<(1,1),(1,1)>$ 
1d

reviewed  Leave Closed Robotics, Cryptography, and Genetics applications of Grothendieck's work? 
1d

reviewed  Close mathematical modelling question 
1d

reviewed  Close Solution to a system of linear equations containing some inequalities 
1d

answered  A perfect $(n,k)$ shuffle function 
1d

comment 
A perfect $(n,k)$ shuffle function
If you label the cards $0$ to $n1$ then your shuffle corresponds to multiplying card $i$ by the inverse of $k$ modulo $n1$. Thus it returns to the original configuration in the order of $k^{1}$ mod $n1$ steps. But this is the same as the order of $k$ mod $n1$. 
1d

comment 
A perfect $(n,k)$ shuffle function
Seems to be the order of $k$ mod $n1$. (This is well known for the usual perfect outer shuffle, and seems numerically to work in your other examples.) 
2d

reviewed  Close Find a prime when some primitive roots are given 
Nov 20 
reviewed  Close “Almost” zeta function 
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reviewed  No Action Needed Normal Covering of a Finite Group 
Nov 19 
reviewed  Leave Open Finite extension of fields with no primitive element 
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reviewed  Close Robotics, Cryptography, and Genetics applications of Grothendieck's work? 
Nov 17 
reviewed  Leave Closed Is there a unique solution? 
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reviewed  Leave Open Piecewise function of a pyramid surface 