Number Theory

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Prime Numbers

solution Prove that there are infinitely many primes.

solution Prove that there are infinitely many primes of the form p=6k-1\,.

solution Prove that the number of primes less than x is bounded below by \log \log x.

solution Prove that there are n\, consecutive composite numbers, for any n>0\,.

solution Prove that any number x\ {\boldsymbol  {\epsilon }}\ {\mathbb  {Z}} can be represented by the sum of Fibonacci numbers.


There are many problems available under Project PEN.

Divisibility

solution Find the remainder when 37^{{100}} is divided by 29.

solution Find the remainder when 45^{{1000}} is divided by 31.

solution Find the remainder when 137^{{153}} is divided by 18.

solution Prove that n^{3}-n is divisible by 6.


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