Legendre symbol- what is the proof that it is a homomorphism?
Thus the Legendre symbol induces a homomorphism from the multiplicative subgroup of $\mathbb{Z}_p$ to its multiplicative subgroup $\{1,-1\}$. Remark: We sketch a proof of the key fact about products of residues and/or non-residues.
Legendre Symbol (2/p) - Mathematics Stack Exchange
Mar 11, 2019 · According to Gauss' lemma, (2 / p) = (− 1)n, where n is the number of integers in the set S = {1 ⋅ 2, 2 ⋅ 2, 3 ⋅ 2,..., (p − 1 2) ⋅ 2} which upon division by p, have remainders greater than p / 2. The members of S are all less than p, so that it suffices to count the number that exceed p / 2. For 1 ≤ k ≤ (p − 1) / 2, we have ...
Proving summations involving the Legendre symbol
Apr 9, 2015 · How can I prove these summations for the legendre symbol How do you find $\sum_{j=0}^{p-1}\left(\frac{j(j+1)}p\right)$,p = prime, where $\left(\frac{j(j+1)}p\right)$ is the Legendre symbol? For the second one, first you can notice that $$\newcommand\jaco[2]{\left(\frac{#1}{#2}\right)}\sum_{a=1}^{p-1} \jaco ap = 0,$$ since this …
Fast legendre symbol calculation - Mathematics Stack Exchange
We know, that the legendre symbol is 1 1 for exactly half of {1, …, p − 1} {1, …, p − 1} and −1 − 1 for the other half, but the way this values are distributed is not clear. The complexity of the usual computation with the laws of quadratic reciprocity is logarithmic in p p. So you could calculate all symbols in O(p log p) O (p log ...
Legendre symbol - Find all primes such that $(7/p)=1$
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Legendre Symbol Sum? - Mathematics Stack Exchange
p. ) =. 0. Where (a p) (a p) represents the Legendre symbol. So far I tried working with Euler’s Criterion, but I haven’t had any luck. I’m feeling like I’m missing something that may be obvious. I’d be grateful for any help offered.
elementary number theory - Legendre's symbol negative sign ...
Nov 8, 2014 · 1 Answer. Sorted by: 4. In general, we can't. What is true is that the Legendre-symbol is multiplicative: (ab p) = (a p)(b p). (a b p) = (a p) (b p). If a = 2 a = 2 and b = −1 b = − 1, (−2 p) =(−1 p)(2 p) (− 2 p) = (− 1 p) (2 p), but this is not necessarily equal to −(2 p) − (2 p). As you can see, it depends on the quadratic ...
Fibonacci Numbers and Legendre symbol - Mathematics Stack …
Apr 5, 2016 · Below is an approach through algebraic number theory, which I hope is acceptable. Let φ = 1 + 2. Then Fn = − − ∈: = [1 + 2]. And we may study the divisibility of Fn in , as the divisibility of two rational integers in is equivalent with the divisibility in . The case p = 5 is either checked by F ≡ (φ − ¯ φ) (mod5) or by direct ...
Using gauss's lemma to find $(\\frac{n}{p})$ (Legendre Symbol)
May 5, 2015 · Gauss' lemma tells us to look at the number of negative least residues in the list of numbers. a, 2a, 3a, ..., $\frac{p-1}{2}a$
Legendre symbol $(-11/p)$ - Mathematics Stack Exchange
May 30, 2016 · Legendre Symbol $\left(\frac4p\right)$ is always congruent to $1$? 5. Proving summations involving the ...