Number Theory and Algebra Problems

NUMBER THEORY AND ALGEBRA PROBLEMS

1. 

2.   Prove/disprove that:
a.    

b.    
   

c.              
      
3.  Given a square of integral side length x, find the maximum number of squares with prime
side lengths that can fit into it, as a function of x: Sp(x).

4.  Find all fractions of the form a/b, where a and b are primes, consisting only of prime
digits.

5.  Find a string of X consecutive composite numbers that cannot be expressed as the sum of
(X+1)! and k, where k is a natural from 2 to X+1. 

6. Consider naturals a, b, and c to be less than a given value N. a, b, and c are arbitrary perfect


powers. If a, b, and c satisfy the triangle inequality, find their maximum values.

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