January+10


 * Problem:** Show that the equation m^2=3 n^2 has no non-zero integer solutions.


 * Problem**: Show that the equation x^2+x=y^2 has no solutions with both x and y non-zero integers.

1) Show that the equation x^2+2y^2=z^2 has no solutions with both x and y odd. 2) Show that if x,y,z have no common factors and satisfy the equation above, then both x and z must be odd while y is even. 3) Show that if x,y,z have no common factors and satisfy the equation above, then one can find integers s,t such that either z-x = 2 s^2, z+x=t^2 or z-x=s^2, z+x=2t^2 4) Solve the equation x^2+2y^2=z^2 in integers (i.e. find a formula that generates all possible solutions).
 * Problem**:


 * Problem**: Find at least one integer solution to the equation 21x - 13 y = 1


 * Problems from textbook:**

A primitive Pythagorean triple is a triple (a,b,c) of integers that have no common divisor and satisfy a^2+b^2=c^2.


 * 2.1**) a) We showed that in any primitive Pythagorean triple (a,b,c) either a or b is even. Use the same sort of argument to show that either a or b must be a multiple of 3.

a) Which odd numbers a can appear in a primitive Pythagorean triple (a,b,c)? b) Which even numbers b can appear in a primitive Pythagorean triple (a,b,c)?
 * 2.3**) Make a conjecture and prove it:


 * 2.4**) (33,56,65) and (16,63,65) are examples of primitive Pythagorean triples with the same c. Find another example of two primitive triples with the same c. Can you find three primitive Pythagorean triples with the same c? Can you find more than three?

c) Find all primitive Pythagorean triples (a,b,c) with c=a+2.
 * 2.6**) b) Find a Pythagorean triple (a,b,c) with c=a+2 and c>1000.