For an odd prime ,
Every odd prime factor of an odd square plus four is one modulo four. The integer is five modulo eight, so at least one prime factor is five modulo eight. Choosing the odd square divisible by any proposed finite list excludes the list and proves infinitude.
For an odd prime ,For , write using a primitive root ; then the left side is , which is one exactly when is even and is a square.
For every odd prime,
If listed all primes congruent to seven modulo eight, then for the odd number is seven modulo eight. Every prime factor is one or seven modulo eight, at least one is seven, and none is among the , a contradiction.
Let count primes at most congruent to modulo eight. Bounding the prime-power divisibility ofabove and its size below gives, for all sufficiently large ,
For an integer matrix and prime , the commuting-matrix binomial theorem and divisibility of for give
For , one has when , and for every .
If , the sum of the least positive quadratic residues below isThe proof uses to compare the sum of all residues with its image under doubling.
For , the congruence has a solution exactly when its discriminant is zero or a quadratic residue modulo .
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The Legendre symbol is a mathematical notation that provides a way to determine if a given integer is a quadratic residue modulo a prime number. Specifically, for an integer \( a \) and a prime \( p \), the Legendre symbol is denoted as: \[ \left( \frac{a}{p} \right) \] It is defined as follows: 1. If \( a \) is congruent to 0 modulo \( p \) (i.e.