Let $p$ be an odd prime number. What is the number $N_p$ of $x\in (\mathbb{Z}/p\mathbb{Z})^{\times}$ for which $-4x$ and $-4x+1$ are quadratic residues modulo $p$? Some computations give me : $$N_3=N_5=0,N_7=1,N_{11}=N_{13}=2,N_{17}=3,N_{19}=4,N_{23}=5,N_{29}=6,N_{31}=7,N_{37}=8$$ $$N_{41}=9, N_{43}=10, N_{47}=11,N_{53}=12,N_{59}=N_{61}=14$$ I couldn't drive any formula from this data in terms of $p$. Any suggestions would be welcome.
2026-03-30 09:00:25.1774861225
The number of $x$ for which $-4x$ and $-4x+1$ are quadratic residues mod some odd prime number
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From your numerical data we obviously have $N_p=\lfloor \frac{p-3}{4}\rfloor$. Can you prove this, using elementary number theory?