Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:Theorem 1. Let x, y, z be positive integers where x is odd and y is even. Then x, y, z is a primitive
Pythagorean triple if and only if there are positive integers u and v such that
v², y= 2uv,
z = u? + v?
(1)
where (u, v) = 1, u > v and exactly one of u and v is even.
Theorem 2 (Euler's Theorem). Let m be a natural number and let a be an integer such that
(a, m) = 1. Then aº(m) = 1 (mod m).
Theorem 3. Let m e N and let a € Z be such that (a, m) = 1. Let n be a natural number. Then
a" = 1 (mod m) if and only if om(a) |n.
(a, b). If d{ c, then the
Theorem 4. Suppose that a and b are non-zero integers and let d =
equation
ax + by = c
(2)
does not have any integer solutions. If d|c, then this equation has an infinite number of solutions.
In fact, if ro and yo is one integer solution to equation (2), then all such solutions have the form
x = xo +t
a
y = Yo – t
where t varies over the integers.
Definition 5. Let m e N and let a e Z be such that (a, m) = 1. The order (mod m) of a,
denoted by om(@), is the least d e N such that ad = 1 (mod m).
Definition 6. Let m be a natural number and let a be an integer such that (a, m)
Om (a) = 0(m), then a is said to be a primitive root (mod m).
= 1. If
Corollary 7. Let m e N. If a primitive root (mod m) exists, then there are exactly o(0(m))
incongruent primitive roots (mod m).
Theorem 8. Let p be a prime number and let d e N be such that d|(p – 1). Then there are
exactly ø(d) many incongruent integers of order d modulo p.
Corollary 9. Let p be a prime number. Then there are exactly ø(p–1) many incongruent integers
of order p – 1 modulo p; that is, there are exactly ø(p – 1) many primitive roots (mod p).
Lemma 10. Let p >1 be a prime number and let a > 1. Then
$(p*) = pª – pª=1 = p° ( 1 –
Theorem 11. Let a, b, n E Z. If n and a are relatively prime and n| (ab), then n|b.
Theorem 12. Let a, b, n be integers where a |n and b|n. If (a, b) = 1, then (ab) |n.
Lemma 13 (Euclid's Lemma). Let a and b be natural numbers and let p be a prime. If p|(ab),
then p|a or p|b.
Corollary 14. Let a be a natural number and p be a prime. If p|a², then p|a.
Theorem 15. Let a, r, m be integers. Then (a, m) = 1 and (r, m) = 1 if and only if (ar, m) = 1.
Theorem 16. Let a, b, m be integers where m > 1. Suppose that the congruence a = b (mod m)
holds. Then (a, m) = 1 if and only if (b, m) = 1.
Corollary 17. Let a, x, b, m be integers where m > 1. If (b, m) = 1 and ax = b (mod m), then
(а, т) — 1 and (х, m) — 1.
![Let
be an odd prime number. Suppose that a E Z is not a
= a has no integer
quadratic residue (mod p). Show that Diophantine equation 4.x4 + py²
solutions.
[You must clearly write and justify your derivation.]](https://content.bartleby.com/qna-images/question/34c061c7-811d-42a4-9c9e-b840019f7270/47b06c98-dd6d-44ac-9e93-faedd4780d70/tshf6s_processed.jpeg)
Transcribed Image Text:Let
be an odd prime number. Suppose that a E Z is not a
= a has no integer
quadratic residue (mod p). Show that Diophantine equation 4.x4 + py²
solutions.
[You must clearly write and justify your derivation.]
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