I was playing around trying to find a function which matches the prime numbers (futile, I know) when I stumbled upon this little thing:(adsbygoogle = window.adsbygoogle || []).push({});

f(n) = n^2 - (n-1)^2 (for n>1, it seems)

which gives

f(2) = 3

f(3) = 5

f(4) = 7

f(5) = 9 (off by -2)

f(6) = 11

f(7) = 13

f(8) = 15 (-2)

f(9) = 17

f(10) = 19

and so on. I only verified this up to n = 32 (at which point you get 32^2 = 1024 and the numbers become cumbersome to work with by hand, for me at least), but it seems like the function f(n) returns prime number n, only with a few false positives thrown in between (at -2 or -4 off).

At least it does not seem to skip any primes :)

I cannot seem to get any obvious patters within the false positives. The gap seem to vary, it is not the case that the function is off only when n itself is prime, and so on. The only thing i can see that the first 32 n implies is that if f(n) is off by -4 then f(n+1) is off by -2 and then f(n+2) = Pn. (well, not Pn, but the next P after f(n-1) at least).

Now I understand that such a simple function which seems so close to the primes must have been extensively studied, so I wondered if anyone knew of any interesting properties here, or perhaps a paper or a book or something.

k

Code (Text):

edit: included below is the table for as far as I went with it, leading zero for readability.

f(02) = 004 - 001 = 3

f(03) = 009 - 004 = 5

f(04) = 016 - 009 = 7

f(05) = 025 - 016 = 9 (-2)

f(06) = 036 - 025 = 11

f(07) = 049 - 036 = 13

f(08) = 064 - 049 = 15 (-2)

f(09) = 081 - 064 = 17

f(10) = 100 - 081 = 19

f(11) = 121 - 100 = 21 (-2)

f(12) = 144 - 121 = 23

f(13) = 169 - 144 = 25 (-4)

f(14) = 196 - 169 = 27 (-2)

f(15) = 225 - 196 = 29

f(16) = 256 - 225 = 31

f(17) = 289 - 256 = 33 (-4)

f(18) = 324 - 289 = 35 (-2)

f(19) = 361 - 324 = 37

f(20) = 400 - 361 = 39 (-2)

f(21) = 441 - 400 = 41

f(22) = 484 - 441 = 43

f(23) = 529 - 484 = 45 (-2)

f(24) = 576 - 529 = 47

f(25) = 625 - 576 = 49 (-4)

f(26) = 676 - 625 = 51 (-2)

f(27) = 729 - 676 = 53

f(28) = 784 - 729 = 55 (-4)

f(29) = 841 - 784 = 57 (-2)

f(30) = 900 - 841 = 59

f(31) = 961 - 900 = 61

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# Primes (sorry)

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