Dumb math fact

[cnoocy]

19² * 4 + 1 = 17² * (4 + 1)

Comments

[thedan.livejournal.com]

23^2 * 5 + 1 = 21^2 * (5 + 1)
27^2 * 6 + 1 = 25^2 * (6 + 1)
31^2 * 7 + 1 = 29^2 * (7 + 1)
35^2 * 8 + 1 = 33^2 * (8 + 1)
39^2 * 9 + 1 = 37^2 * (9 + 1)

and so on...

[cnoocy]

Okay, I can justify that to myself with a bit of polynomial multiplication, but how did you figure out the rule so quickly?

[thedan.livejournal.com]

Started with:

(a^2)b+c = (d^2)(b+c)

rearranged to:

(a^2-d^2)(b+c) = c(a^2-1)

Decided c = 1 would make life easier:

(a^2-d^2)(b+1) = (a^2-1)

Wanted to make the leftmost factor simpler, so I set a = d+2 (based on your example):

(4d+4)(b+1) = d^2+4d+3
4(d+1)(b+1) = (d+1)(d+3)
4(b+1) = d+3
4b+1 = d

And we already know a in terms of d. There's probably a more general relationship I missed by making the assumptions c = 1 and a = d+2, but this is not the math I'm being paid to do. :)

[dougo.livejournal.com]

3² * 0 + 1 = 1² * (0 + 1)

[lorelei-sakai.livejournal.com]

I'm just curious... What were you doing that led you to notice this fact?

[cnoocy]

Um, factoring the time on my cell phone's calculator. [is embarrassed]

[jordanwillow.livejournal.com]

you know, i listen to the odd-numbered symphonies now and then but i hardly ever listen to the even-numbered symphonies. my favorites are 3 and 7, but maybe i need to go back to the evens for a bit.

(that's about as mathematical as my thinking ever gets, btw.)