5.3 Diophantine Equations 279- Prove that the equation
x^3 +y^3 +z^3 +t^3 = 1999has infinitely many integer solutions.812.Prove that for every pair of positive integersmandn, there exists a positive integer
psatisfying
(√
m+√
m− 1 )n=√
p+√
p− 1.5.3.4 Other Diophantine Equations...............................
In conclusion, try your hand at the following Diophantine equations. Any method is
allowed!
813.Find all integer solutions(x, y)to the equation
x^2 + 3 xy+ 4006 (x+y)+ 20032 = 0.814.Prove that there do not exist positive integersxandysuch thatx^2 +xy+y^2 =x^2 y^2.
815.Prove that there are infinitely many quadruplesx, y, z, wof positive integers
such that
x^4 +y^4 +z^4 = 2002 w.816.Find all nonnegative integersx, y, z, wsatisfying
4 x+ 4 y+ 4 z=w^2.817.Prove that the equation
x^2 +y^2 +z^2 + 3 (x+y+z)+ 5 = 0has no solutions in rational numbers.818.Find all positive integersx, ysuch that 7x− 3 y=4.
819.Find all positive integersxsatisfying
32
x!
= 23
x!
+ 1.820.Find all quadruples(u, v, x, y)of positive integers, whereuandvare consecutive
in some order, satisfying
ux−vy= 1.