# Squares and Primes **URL:** \ **Category:** Factual Questions\ **Created:** [March 10, 2015, 1:39am UTC](https://boards.straightdope.com/t/squares-and-primes/714661 "2015-03-10T01:39:57Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [March 10, 2015, 1:39am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/1 "2015-03-10T01:39:57Z")
Fiddling around with my calculator today I found a notable trend in addition of sequential squares: 4+1=5…prime 9+4=13…prime 16+9=25…5^2 25+16=41…prime 36+25=61…prime 49+36=85…5_17 64+49=113…prime 81+64=145…5_29 100+81=181…prime 121+100=221…13_17 144+121=265…5_53 169+144=313…prime 196+169=365…5_73 225+196=421…prime 256+225=481…13_37 289+256=545…5_109 324+289=613…prime 361+324=685…5_137 400+361=761…prime 441+400=841…29^2 Half are prime, and the majority of composites are divisible by 5. Does any of this reveal anything remarkable? ---
**Author:** ![jezzaOZ](https://avatars.discourse-cdn.com/v4/letter/j/a9a28c/32.png) [@jezzaOZ](https://boards.straightdope.com/u/jezzaOZ)\ **Post date:** [March 10, 2015, 1:48am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/2 "2015-03-10T01:48:32Z")
> [@Enola\_Straight](#): > > Does any of this reveal anything remarkable? You have too much time on your hands? ---
**Author:** ![Robert163](https://avatars.discourse-cdn.com/v4/letter/r/4af34b/32.png) [@Robert163](https://boards.straightdope.com/u/Robert163)\ **Post date:** [March 10, 2015, 2:13am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/3 "2015-03-10T02:13:56Z")
sounds interesting 🙂 ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [March 10, 2015, 2:17am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/4 "2015-03-10T02:17:00Z")
What you are noticing is that (2n^2 + 2n + 1) mod 5 = 0 has the solutions n = 5m+1 and n = 5m +3, where m is a whole number. . Or in other words 2/5 of the time the answer is going to be divisible by 5 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 10, 2015, 2:39am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/5 "2015-03-10T02:39:23Z")
If you want to have fun with primes and squares, try this one: The square of every prime number is one more than a multiple of 24. (Obviously except 2 and 3.) ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [March 10, 2015, 2:43am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/6 "2015-03-10T02:43:38Z")
In terms of the prime values they will become less common as n goes to infinity. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 10, 2015, 4:21am UTC](https://boards.straightdope.com/t/squares-and-primes/714661/7 "2015-03-10T04:21:52Z")
> [@](#): > > What you are noticing is that (2n^2 + 2n + 1) mod 5 = 0 has the solutions n = 5m+1 and n = 5m +3, where m is a whole number. And while we’re at it, it has no solutions mod 2 or 3. If something’s not a multiple of 2 or 3, then the most common way for it to fail to be a prime is going to be for it to be a multiple of 5. ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [March 10, 2015, 3:01pm UTC](https://boards.straightdope.com/t/squares-and-primes/714661/8 "2015-03-10T15:01:25Z")
> [@Exapno\_Mapcase](#): > > If you want to have fun with primes and squares, try this one: > > The square of every prime number is one more than a multiple of 24. (Obviously except 2 and 3.) This blows my mind. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [March 10, 2015, 5:33pm UTC](https://boards.straightdope.com/t/squares-and-primes/714661/9 "2015-03-10T17:33:15Z")
> [@Enola\_Straight](#): > > Half are prime, and the majority of composites are divisible by 5. > > Does any of this reveal anything remarkable? The trend that “half are prime” doesn’t persist as you take more such numbers. I ran a quick program to test the primality of the first million such numbers (i.e., up to 1000000[sup]2[/sup] + 1000001[sup]2[/sup]), and found that a little over 10% of them (104,894 of them, to be exact) are prime. > [@Chronos](#): > > And while we’re at it, it has no solutions mod 2 or 3. No solutions mod 7, 11, 19, or 23 either; but 2/13 of the time, the answer will be divisible by 13; 2/17 of the time, it’ll be divisible by 17; 2/29 of the time, it’ll be divisible by 29.