Editing Quadratic reciprocity (section)

===Factoring &nbsp;''n''<sup>2</sup>&nbsp;−&nbsp;5===
Consider the polynomial <math>f(n) = n^2 - 5</math> and its values for <math>n \in \N.</math> The prime factorizations of these values are given as follows:

{| class="wikitable" style="text-align:right;"
|-
! ''n'' !! colspan=2 | {{tmath|f(n)}} ||&nbsp; &nbsp; &nbsp; &nbsp; !! ''n'' !! colspan=2 | {{tmath|f(n)}} ||&nbsp; &nbsp; &nbsp; &nbsp; !! ''n'' !! colspan=2 | {{tmath|f(n)}}
|-
| 1 || −4 || −2<sup>2</sup> || || 16 || 251 || 251 || || 31 || 956 || 2<sup>2</sup>&sdot;239
|-
| 2 || −1 || −1 || || 17 || 284 || 2<sup>2</sup>&sdot;71 || || 32 || 1019 || 1019
|-
| 3 || 4 || 2<sup>2</sup> || || 18 || 319 || 11&sdot;29 || || 33 || 1084 || 2<sup>2</sup>&sdot;271
|-
| 4 || 11 || 11 || || 19 || 356 || 2<sup>2</sup>&sdot;89 || || 34 || 1151 || 1151
|-
| 5 || 20 || 2<sup>2</sup>&sdot;5 || || 20 || 395 || 5&sdot;79 || || 35 || 1220 || 2<sup>2</sup>&sdot;5&sdot;61
|-
| 6 || 31 || 31 || || 21 || 436 || 2<sup>2</sup>&sdot;109 || || 36 || 1291 || 1291
|-
| 7 || 44 || 2<sup>2</sup>&sdot;11 || || 22 || 479 || 479 || || 37 || 1364 || 2<sup>2</sup>&sdot;11&sdot;31
|-
| 8 || 59 || 59 || || 23 || 524 || 2<sup>2</sup>&sdot;131 || || 38 || 1439 || 1439
|-
| 9 || 76 || 2<sup>2</sup>&sdot;19 || || 24 || 571 || 571 || || 39 || 1516 || 2<sup>2</sup>&sdot;379
|-
| 10 || 95 || 5&sdot;19 || || 25 || 620 || 2<sup>2</sup>&sdot;5&sdot;31 || || 40 || 1595 || 5&sdot;11&sdot;29
|-
| 11 || 116 || 2<sup>2</sup>&sdot;29 || || 26 || 671 || 11&sdot;61 || || 41 || 1676 || 2<sup>2</sup>&sdot;419
|-
| 12 || 139 || 139 || || 27 || 724 || 2<sup>2</sup>&sdot;181 || || 42 || 1759 || 1759
|-
| 13 || 164 || 2<sup>2</sup>&sdot;41 || || 28 || 779 || 19&sdot;41 || || 43 || 1844 || 2<sup>2</sup>&sdot;461
|-
| 14 || 191 || 191 || || 29 || 836 || 2<sup>2</sup>&sdot;11&sdot;19 || || 44 || 1931 || 1931
|-
| 15 || 220 || 2<sup>2</sup>&sdot;5&sdot;11 || || 30 || 895 || 5&sdot;179 || || 45 || 2020 || 2<sup>2</sup>&sdot;5&sdot;101
|}

The prime factors <math>p</math> dividing <math>f(n)</math> are <math>p=2,5</math>, and every prime whose final digit is <math>1</math> or <math>9</math>; no primes ending in <math>3</math> or <math>7</math> ever appear. Now, <math>p</math> is a prime factor of some <math>n^2-5</math> whenever <math>n^2 - 5 \equiv 0 \bmod p</math>, i.e. whenever <math>n^2 \equiv 5 \bmod p,</math> i.e. whenever 5 is a quadratic residue modulo <math>p</math>. This happens for <math>p= 2, 5</math> and those primes with <math>p\equiv 1, 4 \bmod 5,</math> and the latter numbers <math>1=(\pm1)^2</math> and <math>4=(\pm2)^2</math> are precisely the quadratic residues modulo <math>5</math>. Therefore, except for <math>p = 2,5</math>, we have that <math>5</math> is a quadratic residue modulo <math>p</math> iff <math>p</math> is a quadratic residue modulo <math>5</math>.

The law of quadratic reciprocity gives a similar characterization of prime divisors of <math>f(n)=n^2 - q</math> for any prime ''q'', which leads to a characterization for any integer <math>q</math>.