Chapters4and8
... • since computational cost is proportional to size, this is faster than working in the full modulus M • Don’t need to read it, just know it helps with some of the computations in RSA public-key scheme Fall 2002 ...
... • since computational cost is proportional to size, this is faster than working in the full modulus M • Don’t need to read it, just know it helps with some of the computations in RSA public-key scheme Fall 2002 ...
Lecture 3
... An integer p > 1 is called a prime if it is not divisible by any integer other than 1, −1, p and −p. Thus p > 1 is a prime if it cannot be written as the product of two smaller positive integers. An integer n > 1 that is not a prime is called composite (the number 1 is considered neither prime, nor ...
... An integer p > 1 is called a prime if it is not divisible by any integer other than 1, −1, p and −p. Thus p > 1 is a prime if it cannot be written as the product of two smaller positive integers. An integer n > 1 that is not a prime is called composite (the number 1 is considered neither prime, nor ...
1 - Columbia Math Department
... pn ≤ kp1 · p2 · · · pn−1 ≤ k 2 (p1 · · · pn−2 )2 ≤ k 4 (p1 · · · pn−3 )4 ≤ . . . ≤ (2k)(2 ) . Clearly we can pick k such that 2k < e. This result shows that, for example, there are at least 2 primes smaller than 100 or that there are at least 3 primes less than 10, 000. This is clearly a horrible un ...
... pn ≤ kp1 · p2 · · · pn−1 ≤ k 2 (p1 · · · pn−2 )2 ≤ k 4 (p1 · · · pn−3 )4 ≤ . . . ≤ (2k)(2 ) . Clearly we can pick k such that 2k < e. This result shows that, for example, there are at least 2 primes smaller than 100 or that there are at least 3 primes less than 10, 000. This is clearly a horrible un ...
Exam II - U.I.U.C. Math
... It follows from the 3 congruences above that 2561 − 2 is divisible by the primes 3, 11 and 17. Hence, by the Fundamental Theorem of Arithmetic, it must be divisible by their product. Therefore 2561 − 2 ≡ 0(mod 561), i..e., 2561 ≡ 2(mod 561). ...
... It follows from the 3 congruences above that 2561 − 2 is divisible by the primes 3, 11 and 17. Hence, by the Fundamental Theorem of Arithmetic, it must be divisible by their product. Therefore 2561 − 2 ≡ 0(mod 561), i..e., 2561 ≡ 2(mod 561). ...
NRF 10 - 3.1 Factors and Multiples of Whole Numbers Where does
... When a number has exactly two factors, 1 and itself, the number is a prime number. Examples: 2, 3, 5, ___, ___, ___, ___, …. Composite Numbers have more than two factors Note: - The number 1 is a special number – neither prime or composite - The number 0 is composite – infinite number of factors Pri ...
... When a number has exactly two factors, 1 and itself, the number is a prime number. Examples: 2, 3, 5, ___, ___, ___, ___, …. Composite Numbers have more than two factors Note: - The number 1 is a special number – neither prime or composite - The number 0 is composite – infinite number of factors Pri ...
A SET OF CONJECTURES ON SMARANDACHE SEQUENCE
... Study some Smarandache P - (partial) - digital subsequences associated to: - Fibonacci numbers (we were not able to find any Fibonacci number verifying a Smarandache type partition, but we could not investigate large numbers; can you? Do you think none of them would belong to a Smarandache F - parti ...
... Study some Smarandache P - (partial) - digital subsequences associated to: - Fibonacci numbers (we were not able to find any Fibonacci number verifying a Smarandache type partition, but we could not investigate large numbers; can you? Do you think none of them would belong to a Smarandache F - parti ...
SOLUTIONS TO HOMEWORK 3 1(a).
... only two exponentials modulo 37. How many exponentials do you need to test for being a primitive root for some prime p? Ans: Again, from the reasoning in part (b), we know a has order dividing p − 1. So we first calculate the prime factorization of p − 1. Say p − 1 = pn1 1 · · · pnk k , where the pi ...
... only two exponentials modulo 37. How many exponentials do you need to test for being a primitive root for some prime p? Ans: Again, from the reasoning in part (b), we know a has order dividing p − 1. So we first calculate the prime factorization of p − 1. Say p − 1 = pn1 1 · · · pnk k , where the pi ...
2.4 Factors Numbers that are multiplied together are called factors
... Note that a number is always a factor of itself because a x 1 = a When two whole numbers, m and n, multiply to get a product, p (m x n = p), then we can say these facts: 1) p is a multiple of m and n 2) p is divisble by m and n 3) m and n divide evenly into p. 4) m and n are factors of p. Example: 1 ...
... Note that a number is always a factor of itself because a x 1 = a When two whole numbers, m and n, multiply to get a product, p (m x n = p), then we can say these facts: 1) p is a multiple of m and n 2) p is divisble by m and n 3) m and n divide evenly into p. 4) m and n are factors of p. Example: 1 ...
PDF9 - Pages
... see if the equality holds. If it fails for any value of , then is composite. If the equality holds for many values of , then we can say that p is probably prime. The presence of Carmichael numbers prevents us from conclusively determining that is a prime. There could be cases where the number ...
... see if the equality holds. If it fails for any value of , then is composite. If the equality holds for many values of , then we can say that p is probably prime. The presence of Carmichael numbers prevents us from conclusively determining that is a prime. There could be cases where the number ...
Lecture 3. Mathematical Induction
... is prime, the exponent n must be prime. This observation goes to the French mathematician of the17th century Mersenne. He claimed first that for all prime P , the number 2P − 1 also is prime, but recognized shortly that it is not always true. It is true for p = 2, 3, 5, 7 but ...
... is prime, the exponent n must be prime. This observation goes to the French mathematician of the17th century Mersenne. He claimed first that for all prime P , the number 2P − 1 also is prime, but recognized shortly that it is not always true. It is true for p = 2, 3, 5, 7 but ...
