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HOMEWORK I THEORY OF NUMBERS

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HOMEWORK I THEORY OF NUMBERS due on September, 14, 2017, by 3.30pm submit by email to pirutka@cims.nyu.edu

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. 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.)

3. Prove or disprove: if a |(b+c), then eithera |b ora|c.

4. Determine the remainder of the division of (a) 2021 by 14;

(b) 2021n+ 1 by 43, where n is a positive integer.

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