# 3) Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that (-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p =+- 1 (mod 8) for odd primes p. # 4) As in #3 but now evaluate (37603|48611), again by regarding it as a Legendre symbol, after first checking by trial division (by hand - be sure to mention how far you need to go to be surel), that 48611 is a prime and 37603 is a semiprime (product of two distinct primes).
# 3) Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that (-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p =+- 1 (mod 8) for odd primes p. # 4) As in #3 but now evaluate (37603|48611), again by regarding it as a Legendre symbol, after first checking by trial division (by hand - be sure to mention how far you need to go to be surel), that 48611 is a prime and 37603 is a semiprime (product of two distinct primes).
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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please just answer 3 & 4. Thank you
Transcribed Image Text:#3) Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that
(-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p == +- 1 (mod 8) for odd primes p.
#4) As in #3 but now evaluate (37603|48611), again by regarding it as a Legendre symbol, after first checking by trial division (by hand - be sure to mention how far you need to go to be sure!), that 48611 is a prime and 37603 is a semiprime (product of two distinct primes).
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