ve squares modulo p) We can use quadratic reciprocity to answer tr question: "How many residue class modulo p are there such that both x and x+1 are squares: (a) Show the identity: р-1 = 0 T=0 IWI
ve squares modulo p) We can use quadratic reciprocity to answer tr question: "How many residue class modulo p are there such that both x and x+1 are squares: (a) Show the identity: р-1 = 0 T=0 IWI
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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Transcribed Image Text:3. (Consecutive squares modulo p) We can use quadratic reciprocity to answer the
question:
"How many residue class modulo p are there such that both x and x+1 are squares?"
(a) Show the identity:
E)-
= 0
x=0
and that if x #0 (mod p)
()
| 1, if y? = x (mod p) is solvable
if y? = x (mod p) has no solutions
1
2
(b) Consider the summation
р-2
(금)
x + 1
1+
N
= 1+ =
1+
+
1+
2
x=1
Justify why N is the number of consecutive pairs of
square
modulo
р.
(c) Prove that
p+2+
4
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