0
$\begingroup$

Can you provide proofs or counterexamples for the claims given below?

First claim

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $M= \frac{k \cdot b^{n}-1}{2} $ where $k$ is an odd positive natural number , $ k<2^n$ , $b$ is an odd positive natural number greater than one and $n\ge3$ . Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{M}\right)=1$ and $\left(\frac{a+2}{M}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol. Let $S_i=P_b(S_{i-1})$ with $S_0$ equal to the modular $P_{kb^2}(a)\phantom{5} \text{mod} \phantom{5} M$. Then, if $M$ is prime then $S_{n-2} \equiv a \pmod{M}$ .

Second claim

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= \frac{k \cdot b^{n}+1}{2} $ where $k$ is an odd positive natural number , $ k<2^n$ , $b$ is an odd positive natural number greater than one and $n\ge3$ . Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{N}\right)=-1$ and $\left(\frac{a+2}{N}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol. Let $S_i=P_b(S_{i-1})$ with $S_0$ equal to the modular $P_{kb^2}(a)\phantom{5} \text{mod} \phantom{5} N$. Then, if $N$ is prime then $S_{n-2} \equiv a \pmod{N}$ .

I have tested these claims for many random values of 𝑘, 𝑏 and 𝑛 and there were no counterexamples.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.