### Two Squares and Four Squares

Diophantus of Alexandria wrote a collection of books called Arithmetica in the 3rd century AD. He noted that natural numbers of the form cannot be expressed as the sum of two square numbers. He also [more…]

Diophantus of Alexandria wrote a collection of books called Arithmetica in the 3rd century AD. He noted that natural numbers of the form cannot be expressed as the sum of two square numbers. He also [more…]

A proof of Fermat’s Little Theorem using necklaces. , where is prime and is any integer with gcd. Multiply both sides of the congruence by : Which gives : Subtract from both sides : [more…]

Euler’s totient function counts the number of positive integers up to that are relatively prime to , where is considered to be relatively prime to all . These numbers are called the totatives of . [more…]

Fermat was able to factorise large numbers, such as the above, long before the days of calculators and computers by making use of his little theorem. One could try trial division by primes less than [more…]

Fermat left only one proof. The area of a Pythagorean triangle is never a square number. Fermat wrote , “If the area of a right-angled triangle were a square, there would exist two biquadrates (fourth [more…]

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