Question

Prove the Basic Principal of Difference of squares: If x2 ≡ y2 (mod n) and x is not ± y, where x and y lie in the range {0, … , n-1}, then n is composite and has gcd(x-y, n) as a non-trivial factor.

Answer #1

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Prove the Basic Principal of Difference of squares: If x2 ≡ y2
(mod n) and x is not ± y, where x and y lie in the range {0, … ,
n-1}, then n is composite
and has gcd(x-y, n) as a non-trivial factor.

Say that x^2 = y^2 mod n, but x != y mod n and x != −y mod
n.
Show that 1 = gcd(x − y, n) implies that n divides x + y, and
that this is not possible, Show that n is non-trivial

topology:
Prove that the open ball B2: = {(x, y) ∈ R2 | x2 + y2 <1} in R2 is homeomorphic to the open squared unit C2: = {(x, y) ∈R2 | 0 <x <1.0 <and <1}

Prove that for n ≥ 5, (n−1)! ≡ 0 mod n if and only if n is
composite. (Take care to consider why your argument would not work
for n ≤ 4. . . )

Prove that there are no rational numbers x and y such that x2
-y2 =1002.

Prove that n is prime iff every linear equation ax ≡ b mod n,
with a ≠ 0 mod n, has a unique solution x mod n.

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

1. Write a proof for all non-zero integers x and y, if there
exist integers n and m such that xn + ym = 1, then gcd(x, y) =
1.
2. Write a proof for all non-zero integers x and y, gcd(x, y) =
1 if and only if gcd(x, y2) = 1.

Solve:
uxx + uyy = 0 in {(x,y) st x2 +
y2 < 1 , x > 0, y > 0}
u = 0 on x=0 and y=0
∂u/∂r = 1 on r=1

A surface x2 +y2 -z = 1 radiates light away. It can be
parametrized as ~r(x; y) = [x, y, x2 + y2 -1]T . Find
the parametrization of the wave front ~r(x,y) + ~n(x, y), which is
distance 1 from the surface. Here ~n is a unit vector normal to the
surface.

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