[PDF] Determination of the Primality of $N$ by Using Factors of $N^2 pm 1$

Number Theory

Show that n = 1 or 2. 8. Let m n be positive integers. Show that 4mn − m Show that if p = 2n + 1

Fermats Little Theorem: Proofs That Fermat Might Have Used

and would have been sufficient to prove that 2P - 2 has a factor p for any a prime number p and 0 < r < n

Public-Key Cryptography

2 янв. 2013 г. Its running time to find a factor p is exp[2 ln(p) lnln(p)]1/2. If n = pq with p and q both near n1/2 then this is L(1) = exp[ln(n) lnln(n)] ...

On Primes Recognizable in Deterministic Polynomial Time - Sergei

And we can show this if p-1 has a factor F exceeding p² with the property II(n−1 p - 1) <. F(a)' p❘n so that F(a) <nloglogn/loga. Hence it is correct ...

On the Number of Elliptic Pseudoprimes

of these points is a 2-division point modulo p for some prime factor p of n

Improved Stage 2 to P1 Factoring Algorithms

6 нояб. 2008 г. It hopes that some prime factor p of N has smooth p−1. It picks b0 ... Bostan shows that multipoint evaluation of a polynomial of degree < n ...

SOLUTIONS TO PROBLEM SET 1 Section 1.3 Exercise 4. We see

+ 1. There is a prime factor p Qn. Suppose p ≤ n; then p n! = n(n−1)(n−2)2 

About the p-paperfolding words

associated to a p-paperfolding sequence. It is known that the number of factors of length n of a 2-paperfolding sequence (i.e. its complexity function) is 

Concentration inequalities.pdf

24 нояб. 2016 г. ... factor v. The sub-Gaussian property implies that Z − EZ has a sub ... t) ≤ e−(n−1)t2/2 . In other words as soon as µ(A) ≥ 1/2

Determination of the Primality of $N$ by Using Factors of $N^2 pm 1$

where q is some prime divisor of R4 depending on p. Proof Let r = r(p) be an order of apparition of p such that rl(N2 + 1); then.

On the Polynomials Related to the Differential Equation

Number Theory

If p is a prime and n is an integer such that p

(4n2 + 1) then p ? 1 (mod 4). Proof. Clearly

so we need only show that p ? 3 (mod 4).

Average Case Error Estimates for the Strong Probable Prime Test

For example we show PIoo

3.2 The Factor Theorem and The Remainder Theorem

The graph suggests that the function has three zeros one of which is x = 2. It's easy to show that f(2) = 0

Correlations for paths in random orientations of G(np) and G(n

https://uu.diva-portal.org/smash/get/diva2:323339/FULLTEXT01.pdf

University of Plymouth

12 fév. 2006 a natural number n i.e.

PUTNAM TRAINING POLYNOMIALS Exercises 1. Find a polynomial

Let n be an even positive integer and let p(x) be an n-degree polynomial such that 2. Factor p(x) + 1. 3. Prove that the sum is the root of a monic ...

On the Numerical Factors of the Arithmetic Forms ?ⁿ

CARMICHAEL NUMBERS AND KORSELTS CRITERION Fermats

(i) n is squarefree and (ii) for every prime p dividing n also (p ? 1)

Solutions to the Number Theory Problems - Mathematics

We have 2n= p k+1 = (p+1)(pk 1 p 2 + 1) so (p+1) is a power of 2 say p+1 = 2 l Then 2n= p k+1 = (2 l1) +1 = (2l)k (2l)k 1+ +(2 ) 1 +1 = (2l)k (2l) 1+ +(2l) and this is an odd multiple of 2 l Since 2n is an odd multiple of 2 we must have 2n = 2l So we have 2n = p k+ 1 = (2n 1) + 1 This only happens when n= 1 or when k= 1 neither of which

Solutions to the Number Theory Problems - Mathematics

n p has a prime factor q with q < r n p < 3 p n < p and this prime factor q is also a divisor of n; which contradicts the de nition of p: Therefore n p must be prime Question 6 [p 87 #12] Show that every integer greater than 11 is the sum of two composite integers Solution: If n > 11 and n is even then n 4 is even and n 4 > 7; so that n 4

18781 Homework 8 - Massachusetts Institute of Technology

that there is at least one prime factor pof the form 8n+ 5 If not a= Q q i i 1(mod 8) (as a product of numbers congruent to 1(mod 8) is still 1(mod 8)) But (2p 1p 2:::p k)2 4(mod 8) as p2 j 1(mod 8) for every j and a= (2p 1p 2:::p k)2 + 1 5(mod 8) so this is a contradiction At least one of the prime factors of a say p must be of the

Mathematics 4: Number Theory Problem Sheet 4 Workshop 9 Nov 2012

(11) Show that if 2n ?1 is prime then n is prime For if n = pq say with pq > 1 then since y ? 1 divides yq ? 1 we have (y = xp) that xp ?1 divides xn ?1 = (xp)q ?1 Hence (x = 2) 2p ?1 divides 2n ?1 and 2p ? 1 6= 1 6= 2 n ? 1 (12) Show that if 2n +1 is prime then n is a power of 2 For suppose n = m? with ? > 1

FFermat Euler Wilson Linear Congruences Lecture 4 Notes

show that second equals {z factor the ?rst p} 1 (1)1 (mod p) p 2 (1)2 (mod p) p+ 1 p 1 (1) (mod p) 2 2 p+ 1 p1 p 1:::(p 1) ( 1) 2 1 2::: 2 second {z factor} 2 (mod p) {z x} p1 2 is even since p 1 mod 4 and so second factor equals the ?rst factor so x= p1! 2 solves x 2 1 mod pif p 1 mod 4 Theorem 23 There are in?nitely

Searches related to show that 2^p+1 is a factor of n filetype:pdf

If n ‚ 2 and an ¡1 is prime we call an ¡1 a Mersenne prime For which integers a can an ¡1 be prime? We take n ‚ 2 as if n = 1 then a is just one more than a prime We know using the geometric series that an ¡1 = (a¡1)(an¡1 +an¡2 +¢¢¢ +a+1): (1) So a¡1 j an¡1and therefore an¡1will be composite unless a¡1 = 1 or

What is P Q1 1 n 1 n k?

p q1 1 n 1 n k decrease with each value of k. Eventually, the numerator becomes zero, and we obtain p q= 1 n 1 +1 n 2 + +1 n

How do you factor 2n1?

Also, if n is composite, so that n = k ‘; with k > 1 and ‘ > 1; then we can factor 2n1 as in the hint: 2k‘1 = (2k1)(2k(‘ 1)+2k(‘ 2)+ +2k+1): and each factor on the right is clearly greater than 1: which is a contradiction, so n must be prime. Question 3.

Which integer of the form n3+1 is a prime?

Question 1. [p 74. #6] Show that no integer of the form n3+1 is a prime, other than 2 = 13+1: Solution: If n3+1 is a prime, since n3+1 = (n+1)(n2n+1); then either n +1 = 1 or n2n+1 = 1: The n +1 = 1 is impossible, since n  1; and therefore we must have n2n+1 = 1; that is, n(n 1) = 0; so that n = 1: Question 2.

What is the smallest prime factor of N?

p n; then n p must be prime or 1: Solution: Let p be the smallest prime factor of n; and assume that p >3 p n: Case 1: If n is prime, then the smallest prime factor of n is p = n; and in this case n p = 1: Case 2: If n > 1 is not prime, then n must be composite, so that n = p n p ; and since p >3