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High School
By
Shiva Kintali
on June 2, 2013 | Updated Dec. 6, 2017
Prime magic
Let \(p\) be a prime number bigger than 3. Prove that \(p^2-1\) is always divisible by \(24\).
Mathematics
Number Theory
primes
0
High School
By
Shiva Kintali
on Dec. 14, 2013 | Updated Jan. 4, 2018
Legendre's Theorem
Prove the following Legendre's Theorem : Legendre's Theorem : The number \(n!\) contains the prime factor \(p\) exactly \(\sum_{k \geq 1}{\lfloor \frac{n}{p^k} \rfloor}\) times.
Mathematics
Number Theory
primes
0
High School
By
Shiva Kintali
on Dec. 31, 2013 | Updated Dec. 6, 2017
Prime gaps are not bounded
Prove that the gap between consecutive primes is not bounded, by proving the following theorem. Given any integer \(k \geq 1\), there is a number \(N\) such that \(N+1, N+2, N+3, \dots, N+k\) are …
Mathematics
Number Theory
primes
0
High School
By
Shiva Kintali
on March 24, 2013 | Updated Dec. 6, 2017
Infinite number of special primes
Prove that there are infinitely many primes of the form \(4n-1\) and \(4n+1\).
Mathematics
Number Theory
primes
0
High School
By
True IMO
on Oct. 9, 2012 | Updated Jan. 4, 2018
International Mathematical Olympiad 1998 Problem 3
For any positive integer \(n\), let \(d(n)\) denote the number of positive divisors of \(n\) (including \(1\) and \(n\) itself). Determine all positive integers \(k\) such that \(d(n^2)/d(n) = k\) for…
Mathematics
Number Theory
divisibility
imo
imo 1998
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