Question

Number theory: Explain why n^2 - n + 41 is a prime number for all integer values of n between 0 and 40. (In fact, between -39 and 40!)

Answer 1

off the top of my head i would say it is because you cannot factor \(n^2-n+41\)

Answer 2

That's true, but if you try others like \[n ^{2}-n+29\] it doesn't work.

Answer 3

then i am wrong

Answer 4

hmm you can look here
http://en.wikipedia.org/wiki/Formula_for_primes

Answer 5

here is an actual proof
http://planetmath.org/encyclopedia/ProofThatN2N41IsPrimeFor0leqNleq39.html
maybe you can explain what this line means

"Recall that there is only one reduced integral binary quadratic form of discriminant"

Answer 6

pretty interesting problem. i think it will take me much longer to understand the proof than it will to verify that it is true with a calculator, but if this makes sense to you and you explain what the first line means, i can follow the logic from there

Answer 7

Thanks for the two links, I followed through to the explanation of that line ("Recall that there is only one reduced integral binary quadratic form of discriminant") here: http://planetmath.org/encyclopedia/IntegralBinaryQuadraticForms.html and I just about managed upto where the matrices begin. After that, though, they start talking about "GL"-something but I think at that point it's kind of beyond my level :D Thanks again though, it was a thoroughly good read, as far as I managed :)

Answer 8

Interestingly (but off topic), the Wikipedia page you mentioned had a link to the Ulam Spiral http://en.wikipedia.org/wiki/Ulam_spiral , and I just realised it's the same spiral I made in school a couple of weeks ago, while bored in class :) Maybe, if I was born 50 years earlier, I could have discovered it first!

Answer 9

yeah general linear something, square matrices with non - zero determinants, but i don't think that is the essential part
i am wondering what you were supposed to do for this exactly.

Answer 10

number theory is realy the best! i just wish i was smarter.

Answer 11

if you come up with an answer for this please come back and post. i am imagining that this is a homework exercise, so maybe you will see a solution
is this elementary number theory class or something else?

Answer 12

wow it gets more interesting. heegner numbers? check this out
http://en.wikipedia.org/wiki/Heegner_number
i have to run

Answer 13

Yeah, it's a homework excercise, we started number theory about a month ago in school but it's pretty tough :) The actual homework problem only asked to check that it held true, using a calculator, but then it asks for a proof that \[n ^{2}-3961n+3922339\] is prime, where "n is the year of your birth" :D I thought this would be simpler to prove first though, before trying the other one.

Answer 14

I'll check it out, thanks for all your help :)