Sometimes i like to waste my time observing the behavior of certain functions and identify any patterns. Most of the time i just look at the differences (absolute value) between two consecutive elements of a sequence of numbers and look for a pattern. If no pattern is evident i take the difference of the differences and look for a pattern again, and so on.(adsbygoogle = window.adsbygoogle || []).push({});

This creates a triangle of differences. Recently i did a program that does all of this work automatically and so i had some fun generating the triangle for various functions.

I did it for the primes and found no pattern, naturally, save for some scattered n-sized triangles of 0s, bounded by 2s, like the following:

http://www.bloo.us/temp/primes.aspx?id=0 [Broken]

I also tried an exponential function, 2^n and obtained a cool triangle:

http://www.bloo.us/temp/primes.aspx?id=6 [Broken]

But i found really interesting was the polynomial functions. Granted i'm not a mathematician, but i found the following result surprising:

- for any polynomial of the form ax^d + bx^(d-1) ... cx^0, after d+1 rows all the differences become 0. This means that the (d+1)th differences are all the same. Hard to explain, easier to look at it:

For x^2:

http://www.bloo.us/temp/primes.aspx?id=1 [Broken]

Differences at d+1 row are all 2

For x^3:

http://www.bloo.us/temp/primes.aspx?id=2 [Broken]

Differences at d+1 row are all 6

For x^4:

http://www.bloo.us/temp/primes.aspx?id=3 [Broken]

Differences at d+1 row are all 24

For 23x^3 + 5x^2:

http://www.bloo.us/temp/primes.aspx?id=4 [Broken]

Differences at d+1 row are all 138

For 6x^4 + 11x^3 + 4x^2:

http://www.bloo.us/temp/primes.aspx?id=5 [Broken]

Differences at d+1 row are all 144

What seems to follow is that there's no polynomial that produces all the primes. There's no polynomial that produces the first 200 primes either. Though i can, from looking at the primes' triangle see that there is a degree 2 polynomial that produces:

41 43 47 53

And another degree 2 polynomial that produces:

673 677 683 691 701

etc...

But most of it is non-polynomial. The problem is that the differences of a polynomial are either increasing or the same, there's never any zeros and the primes' triangle is filled with 0s.

I have the triangle for the first 200 primes:

http://www.bloo.us/temp/primes200.htm [Broken]

It's pretty big, about 2Mb.

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# Something i didn't know about polynomials

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