3.15. Use the Miller-Rabin test on each of the following numbers. In each case, either provide a Miller-Rabin witness for the compositeness of n, or conclude that n is probably prime by providing 10 numbers that are not Miller-Rabin witnesses for n. n (c) n = 294439 (a) n = 1105. (Yes, 5 divides n, but this is just a warm-up exercise!) (b) n = 294409 (d) n = 118901509 (f) n = 118901527 (e) n = 118901521 118915387 Solution to Exercise 3.15. (a) n − 1 = 1104 = 24. 69. 60 = n = 269 = -138 (mod 1105) (mod 1105) (mod 1105) (mod 1105) Thus 1105 is composite. It factors as n = 5.13.17. 22.69 = 259 24-69-324 28-69 1
3.15. Use the Miller-Rabin test on each of the following numbers. In each case, either provide a Miller-Rabin witness for the compositeness of n, or conclude that n is probably prime by providing 10 numbers that are not Miller-Rabin witnesses for n. n (c) n = 294439 (a) n = 1105. (Yes, 5 divides n, but this is just a warm-up exercise!) (b) n = 294409 (d) n = 118901509 (f) n = 118901527 (e) n = 118901521 118915387 Solution to Exercise 3.15. (a) n − 1 = 1104 = 24. 69. 60 = n = 269 = -138 (mod 1105) (mod 1105) (mod 1105) (mod 1105) Thus 1105 is composite. It factors as n = 5.13.17. 22.69 = 259 24-69-324 28-69 1
Chapter9: Sequences, Probability And Counting Theory
Section9.5: Counting Principles
Problem 4SE: Answer the following questions. 4. Describe how the permutation of ii objects differs from the...
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Transcribed Image Text:3.15. Use the Miller-Rabin test on each of the following numbers. In each
case, either provide a Miller-Rabin witness for the compositeness of n, or
conclude that n is probably prime by providing 10 numbers that are not
Miller-Rabin witnesses for n.
n
(c) n = 294439
(a) n = 1105. (Yes, 5 divides n, but this is just a warm-up exercise!)
(b) n = 294409
(d) n = 118901509
(f) n = 118901527
(e) n = 118901521
118915387
Solution to Exercise 3.15.
(a) n − 1 = 1104 = 24. 69.
60
=
n =
269 = -138
(mod 1105)
(mod 1105)
(mod 1105)
(mod 1105)
Thus 1105 is composite. It factors as n = 5.13.17.
22.69 = 259
24-69-324
28-69 1
Transcribed Image Text:2.
For each of the following numbers n, provide a Miller-Rabin witness for the compositeness of
n. (a) 15; (b) 21; (c) 25; (d) 35.
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