4. (i) For an odd number n, suppose that 2" # 2 mod n. Can n be a prime? Explain. This is called the base 2 primality test. (ii) Using the fact that 187 = 11-17, show that 2186174 mod 187. This shows that 187 fails the base 2 primality test. (iii) Compute 2561 mod 561. Use Fermat's little theorem and hand calculations. Does 561 pass or fail the base 2 primality test? You may use the prime factorization of 561.
4. (i) For an odd number n, suppose that 2" # 2 mod n. Can n be a prime? Explain. This is called the base 2 primality test. (ii) Using the fact that 187 = 11-17, show that 2186174 mod 187. This shows that 187 fails the base 2 primality test. (iii) Compute 2561 mod 561. Use Fermat's little theorem and hand calculations. Does 561 pass or fail the base 2 primality test? You may use the prime factorization of 561.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Number Theory] How do you solve question 4? thanks
Transcribed Image Text:1. Find all solutions of the congruence
5x² + 3x² + x + 3 = 0 mod 30.
You should do this by factoring the modulus and using the CRT, as explained in class.
2. Recall that a number m is said to be square free if d² | m for d≥ 1 implies that d = 1.
Equivalently, m is not divisible by the square of any prime p. Show that there are infinitely
many integers n such that each of the numbers n, n+1, n+2 and n+ 3 not square free.
3. Solve the system of congruences
3x2 mod 7
7x3
mod 11
8x 7
mod 19.
Show the steps used.
4. (i) For an odd number n, suppose that
2" # 2 mod n.
Can n be a prime? Explain. This is called the base 2 primality test.
(ii) Using the fact that 187 = 11-17, show that
2186174 mod 187.
This shows that 187 fails the base 2 primality test.
(iii) Compute
2561 mod 561.
Use Fermat's little theorem and hand calculations.
Does 561 pass or fail the base 2 primality test?
You may use the prime factorization of 561.
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