Assignment 4, Math 313
Due: Wednesday, March 30th, 2016
1 Prove that the Diophantine equation x
4 − y
4
= z
2 has no solution in nonzero integers x, y and z . Hint : parallel the proof via infinite descent used for the equation x
4
+ y
4
= z
2
.
2 Prove that the Diophantine equation x
4 − y
4
= 2 z
2 has no solution in nonzero integers x, y and z . Hint : factor x 4 and search for a primitive Pythagorean triple.
− y 4
3 Determine all integers x and y such that
| 2 x − 3 y | = 1 .
This was first done roughly 700 years ago!
4 Show that the equation y
2
= x
3
+ 7 has no solutions in integers. Hint : add 1 to both sides of the equation and factor the right hand side. Then consider things modulo 4.
5 Prove that a positive integer is expressible as the difference of two squares of integers if and only if it is not congruent to 2 modulo 4.
6 Let n ≥ 170 be an integer. Prove that n is expressible as the sum of five squares of positive integers. Hint : express n = 169 as the sum of four squares of nonnegative integers, say n − 169 = a
2
+ b
2
+ c
2
+ d
2
.
If none of these squares is zero, then n = a 2 + b 2 + c one of the squares is zero, say d = 0, then n = a
2
2 + d 2
+ b
2
+13 2 . If exactly
+ c
2
+ 5
2
+ 13
2
.
Continue.
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