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Number Theory
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Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Primes and divisibility
Prove the following without using Fermat's little theorem or Euler's totient theorem. Here \(a\ |\ b\) means \(a\) divides \(b\). Prove that for every prime number \(p\) and every pair of integer…
Mathematics
Number Theory
divisibility
primes
0
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
Undergraduate
By
True Putnam
on June 7, 2012 | Updated Jan. 4, 2018
Putnam 2005 A1
Prove that every positive integer can be expressed as a sum of numbers of the kind \(2^i3^j\), where no term in the summation divides the other. For example : 1) 14 = 8 + 6 is a valid represent…
Computer Science
Mathematics
Algorithms
Number Theory
dynamic programming
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Undergraduate
By
Shiva Kintali
on Oct. 12, 2013 | Updated Jan. 4, 2018
Ramsey primes
For every integer \(m \geq 1\), there exists an integer \(p_0\) such that, for all primes \(p \geq p_0\), the congruence \(x^m + y^m \equiv z^m (\mbox{mod}\ p)\) has a solution with positive …
Mathematics
Number Theory
primes
ramsey theory
0
Graduate
By
Shiva Kintali
on June 12, 2012 | Updated Dec. 6, 2017
Solving Discrete-logarithm
Consider the following problem : Given a \(y\) such that \(0 < y < p\), where \(p\) is a prime number, find an \(x\) (if it exists) such that \(2^x ≡ y\ \mbox{mod}\ p\). Let \(n\) be the number …
Computer Science
Mathematics
Algorithms
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
Graduate
By
Shiva Kintali
on June 19, 2012 | Updated Dec. 6, 2017
Primality is in NP $\cap$ co-NP
Primality is the following problem : Given a positive integer \(n\), is \(n\) prime ? Note that the size of the input is the number of bits used to represent \(n\). Easy : Show that Primality…
Computer Science
Mathematics
Complexity Theory
Number Theory
primes
0
Graduate
By
Shiva Kintali
on Aug. 13, 2012 | Updated Dec. 6, 2017
Integer Factoring is in NP $\cap$ co-NP
Formulate an appropriate decision version of Integer Factorization and prove that it is in NP \(\cap\) co-NP. There are two ways of proving this : (Easy) Use the fact that Primality is in P. Witho…
Computer Science
Mathematics
Complexity Theory
Number Theory
factoring
NP completeness
0
Undergraduate
By
Shiva Kintali
on Dec. 14, 2013 | Updated Jan. 4, 2018
Product of Primes
Prove that \(\displaystyle \prod_{p \leq x} p \leq 4^{x-1}\) for all real \(x \geq 2\). Here the product is taken over all prime numbers \(p \leq x\).
Mathematics
Number Theory
primes
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Primes and Combinations
Prove that if \(p\) is prime and \(0 < k < p\), then \(p\ |\ {p \choose k}\). Conclude that for all integers \(a\) and \(b\) and all primes \(p\), \((a+b)^p \equiv a^p + b^p\ (\mbox{mod}\ p)\) Prove…
Mathematics
Number Theory
primes
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