Difference between revisions of "Number Theory"

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[[NT1|solution]] Prove the existence of infinitely many primes.
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== Prime Numbers ==
  
[[NT2|solution]] Prove that the number of primes below <math>x</math> is bounded below by <math>\log\log x</math>.
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[[NT1|solution]] Prove that there are infinitely many [[prime|primes]].
  
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[[NT8|solution]] Prove that there are infinitely many primes of the form <math>p=6k-1\,</math>.
  
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[[NT2|solution]] Prove that the number of primes less than <math>x</math> is bounded below by <math>\log\log x</math>.
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[[NT3|solution]] Prove that there are <math>n\,</math> consecutive [[composite]] numbers, for any <math>n > 0\,</math>.
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[[NT9|solution]] Prove that any number <math> x\ \boldsymbol{\epsilon}\  \mathbb{Z} </math> can be represented by the sum of [[Fibonacci numbers]].
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There are many problems available under [http://www.mathlinks.ro/index.php?f=456 Project PEN].
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== Divisibility ==
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[[NT4|solution]] Find the remainder when <math>37^{100}</math> is divided by 29.
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[[NT5|solution]] Find the remainder when <math>45^{1000}</math> is divided by 31.
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[[NT6|solution]] Find the remainder when <math>137^{153}</math> is divided by 18.
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[[NT7|solution]] Prove that <math>n^3 - n</math> is divisible by 6.
  
  
  
 
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Latest revision as of 05:31, 14 March 2009

Prime Numbers

solution Prove that there are infinitely many primes.

solution Prove that there are infinitely many primes of the form .

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

solution Prove that there are consecutive composite numbers, for any .

solution Prove that any number can be represented by the sum of Fibonacci numbers.


There are many problems available under Project PEN.

Divisibility

solution Find the remainder when is divided by 29.

solution Find the remainder when is divided by 31.

solution Find the remainder when is divided by 18.

solution Prove that is divisible by 6.


Main Page