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HOMEWORK I MATH-UA 0248-001 THEORY OF NUMBERS due on September, 15, 2017 1. It is generally believed that infinitely many primes have the form N

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HOMEWORK I

MATH-UA 0248-001 THEORY OF NUMBERS due on September, 15, 2017

1. It is generally believed that infinitely many primes have the form N2 + 1, although no one knows for sure. Do you think there are infinitely many primes of the form N2−1?

2. Show that for a Pythagorean triple (X, Y, Z) exactly one of the following holds:

(a) all X, Y, Z are multiples of 3;

(b) X is a multiple of 3 and Y is not;

(c) Y is a multiple of 3 and X is not.

3. Find a formula for all points on the hyperbola x2−y2 = 1 whose coordinates are rational numbers. (Hint. Take the line through the point (−1,0) having rational slope k and find a formula in terms of k for the second point where the line intersects the hyperbola.)

4. Fermat’s Last Theorem states that if n∈N and n ≥3, the equation Xn+Yn=Zn

has no solutions in nonzero integers X, Y, Z. Make use of this theorem to prove that √n

2∈/ Qwhere n ∈Nand n ≥3.

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