This practice set will be devoted to seeing an array of algebraic tricks that, when combined with the properties of integers, allows to solve some diophantine equations.
Pure divisibility
Problem 1
Show that the number
cannot be prime for any .
Enumerating remainders
Recall the following lemma:
Lemma
For relatively prime
and (that is, ) if and , then .
Problem 1
Prove that
is divisible by for all integers .
Problem 2
Prove that the number
is divisible by for all integers .
Problem 3 (square remainders)
Can you enumerate the possible remainders that squares of integer numbers can give when divided by: (a)
; (b) ; (c) ; (d) ; (e) ?
Problem 4 (cube remainders)
Can you enumerate the possible remainders that cubes of integer numbers can give when divided by: (a)
; (b) ; (c) ; (d) ?
Problem 5
Show that a square cannot end in two odd digits.
Simon’s favorite factoring trick (SFFT) and factorization
Problem 1
Solve the equation
in integers. Hint:
.
Problem 2
How many integer solutions does the equation
have? What about ?
Problem 3
Find all solutions of the equation
in integers satisfying the additional restriction that and are prime.
Problem 4
Find all solutions to the equation
in prime numbers ( , and are again restricted to be prime.
Problem 5
Consider the equation
Show that, for any prime
, the equation has exactly three solutions in natural numbers, and if is not prime (but still natural), then there are more solutions.
Problem 6
Find all integer solutions to the equation
.
Taking the equation/expression modulo something
Problem 1
Prove that the equation
has no integer solutions. However, the equation has solutions: can you find them?
Problem 2
Show that the equation
has no integer solutions.
Problem 3
Show that there are infinitely many integers that cannot be represented as: (a)
; (b)
; (c)
.
Problem 4
Find all natural
such that the number is a perfect square.
Problem 5
Prove that the equation
has no integer solutions.