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Results tagged with prime-numbers
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user 88804
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
1
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0
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93
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Primality test for specific class of $N=12k \cdot 5^n-1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 12k \cdot 5^{n} - 1 $ where $n\ge3$ , $12k <5^n$ a …
- 2,661
4
votes
1
answer
321
views
Primality test for $N=2^mp^n +1$
This question is related to my previous question.
Can you prove or disprove the following claim:
Let $N=2^mp^n+1$ , $m>0 , n>0$ and $p$ is an odd prime . If there exists an integer $a$ such that $$\d …
- 2,661
1
vote
0
answers
47
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What is the most efficient algorithm for calculating $\Phi_q(b) \operatorname{mod} N$?
From Hurwitz's theorem about irreducible factor $F_{n-1}(x)$ of degree $\varphi(n-1)$ of $x^{n-1}-1$ we can deduce the following criterion for the primality of $N=2^m \cdot p_1^{n_1} \cdot p_2^{n_2} \ …
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0
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0
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123
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Testing the primality of Mersenne and Fermat numbers using third order recurrence relation
Can you prove or disprove the claims given below?
Inspired by generalization of Lucas-Lehmer test I have formulated the following claims:
Claim 1 Let $M_p=2^p-1$ where $p$ is an odd prime number , le …
- 2,661
7
votes
1
answer
460
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Primality test for $N=2^a3^b+1$
Can you prove or disprove the following claim:
Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime.
You can r …
- 2,661
6
votes
1
answer
196
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Primality test for $N=4p+1$
Can you prove or disprove the following claim:
Let $N=4p+1$ where $p$ is an odd prime number , let $T_n(x)$ be the nth Chebyshev polynomial of the first kind and let $F_n(x)$ denote an irreducible fa …
- 2,661
5
votes
1
answer
414
views
Primality test for $2p+1$
In 1750 Euler stated following theorem :
Let $p \equiv 3 \pmod 4$ be prime then $2p+1$ is prime iff $2p+1 \mid 2^p-1$ .
In 1775 Lagrange gave a proof of the theorem .
Recently I have formulated …
- 2,661
4
votes
1
answer
824
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Primality test similar to the AKS test
Let us define polynomials $P_n^{(a)}(x)$ as follows :
$P_n^{(a)}(x)=\left(\frac{1}{2}\right)\cdot\left(\left(x-\sqrt{x^2+a}\right)^n+\left(x+\sqrt{x^2+a}\right)^n\right)$
We can define these polynom …
- 2,661
4
votes
1
answer
110
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Primality test for specific class of natural numbers using factors of Lucas polynomials
This question is related to my previous question.
Can you prove or disprove the following claim:
Let $N=2n+1$ where $n$ is an odd natural number greater than one , let $L_m(x)$ be the mth Lucas polyn …
- 2,661
1
vote
0
answers
116
views
A primality criterion for specific class of $N=4kp^n+1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 4kp^n+1 $ such that $p$ is a prime number greater tha …
- 2,661
3
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1
answer
383
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Primality test for specific class of $N=k \cdot b^n-1$
This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ .
Can you provide a proof or a counterexample to the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left …
- 2,661
2
votes
1
answer
833
views
Primality test for generalized Fermat numbers
This question is successor of Primality test for specific class of generalized Fermat numbers .
Can you provide a proof or a counterexample for the claim given below?
Inspired by Lucas–Lehmer–Riesel …
- 2,661
4
votes
1
answer
182
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Primality test for specific class of $N=8k \cdot 3^n-1$
This question is related to my previous question.
Can you prove or disprove the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
…
- 2,661
10
votes
0
answers
631
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Primality testing using Chebyshev polynomials
Can you provide a proof or a counterexample for the claim given below?
Inspired by an alternative definition of the Frobenius primality test which is given in this paper I have formulated the followin …
- 2,661
5
votes
0
answers
586
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Primality test for specific class of generalized Fermat numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $F_{p,n}= (2p)^{2^n}+1 $ where $p$ is a prime number greate …
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