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Let $$ F_{n} = \frac{1}{\sqrt{5}} \left[ \left( \frac{1+\sqrt{5}}{2}\right)^{n} - \left( \frac{1-\sqrt{5}}{2} \right)^{n} \right] $$ be a Fibonacci number. If $p\neq 2, 5$ is a prime, then I want to show that $$ F_{p} \equiv \left( \frac{p}{5}\right) \mathrm{mod }\,p , $$ where $\left( \frac{p}{5}\right)$ is understood as a Legendre symbol.

My solution is following: I just expand $F_{n}$ using the binomial theorem and get $$ F_{p} \equiv 2^{p-1}F_{p} \equiv \binom{p}{1} + \binom{p}{3}\cdot 5 + \cdots + 5^{(p-1)/2} \equiv 5^{(p-1)/2} \equiv \left( \frac{5}{p} \right) \equiv \left( \frac{p}{5}\right) \mathrm{mod}\,p $$ Here I used quadratic reciprocity in the last equality. I want to know if there's any alternative proof that does not use quadratic reciprocity and show the congruence directly.

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  • $\begingroup$ See Exercise 2.26 in "Reciprocity Laws. From Euler to Eisenstein". Google books displays the page 73. $\endgroup$ – franz lemmermeyer Mar 26 at 14:52
  • $\begingroup$ @franzlemmermeyer Thanks! Is there any generalization of the proof? Actually, I hope that one can find a family of sequences that can be used to prove quadratic reciprocity. I want to know if this kind of proof is already known or not. $\endgroup$ – Seewoo Lee Mar 26 at 14:58
  • $\begingroup$ Robin Chapman found such a proof, which essentially boiled down to computing quadratic Gauss sums, so he did not publish it. $\endgroup$ – franz lemmermeyer Mar 26 at 16:34
  • $\begingroup$ @franzlemmermeyer That's sad...anyway thank you very much. $\endgroup$ – Seewoo Lee Mar 26 at 16:52
  • $\begingroup$ Gauss, who discovered the Quadratic Reciprocity Law, gave 5 or 6 different proofs of it. $\endgroup$ – DanielWainfleet Mar 27 at 4:55
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I'm just unwinding the proof of quadratic reciprocity using cyclotomic fields.

Let $\zeta$ be a primitive $5$-th root of unity, which is a root of the polynomial $1+x+x^2+x^3+x^4$. The Fibonacci numbers satisfy $$ F_n = \left(\frac{1}{5}+\frac{2}{5}\big(\zeta+\zeta^{-1}\big)\right)\bigg[\big(1+\zeta+\zeta^{4}\big)^n-\big(1+\zeta^2+\zeta^3\big)^n\bigg], $$ which is straightforward to prove by induction. I will write $\equiv_p$ for congruence modulo $p$. Using the identity $(x+y)^p\equiv_p x^p+y^p$, we find $$ F_p\equiv_p \left(\frac{1}{5}+\frac{2}{5}\big(\zeta+\zeta^{-1}\big)\right)\bigg[\zeta^p+\zeta^{4p}-\zeta^{2p}-\zeta^{3p}\bigg]. $$ Since $\zeta^5=1$, the value of $\zeta^{k}$ depends only on the residue of $k$ mod $5$. If $p\equiv 1$, $4$ mod $5$, then $$ F_p\equiv_p \left(\frac{1}{5}+\frac{2}{5}\big(\zeta+\zeta^{-1}\big)\right)\bigg[\zeta+\zeta^{4}-\zeta^{2}-\zeta^{3}\bigg]= F_1= 1, $$ while if $p\equiv 2$, $3$ mod $5$, then $$ F_p\equiv_p \left(\frac{1}{5}+\frac{2}{5}\big(\zeta+\zeta^{-1}\big)\right)\bigg[\zeta^2+\zeta^{3}-\zeta-\zeta^{4}\bigg]= -F_1= -1. $$

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