wu :: forums « wu :: forums - sum of the squares of three consecutive odd prime » Welcome, Guest. Please Login or Register. Jul 1st, 2022, 6:00am RIDDLES SITE WRITE MATH! Home Help Search Members Login Register wu :: forums  riddles  medium (Moderators: Grimbal, towr, william wu, ThudnBlunder, Icarus, Eigenray, SMQ)  sum of the squares of three consecutive odd prime « Previous topic | Next topic » Author Topic: sum of the squares of three consecutive odd prime  (Read 1425 times)
Christine
Full Member    Posts: 159 sum of the squares of three consecutive odd prime   « on: Jan 21st, 2017, 10:54am » Quote Modify

83 is a prime number and 83 = 3^2 + 5^2 + 7^2

Can you prove that there are no other prime numbers which are the sum of the squares of the three consecutive odd primes? IP Logged
Uberpuzzler      Posts: 1026 Re: sum of the squares of three consecutive odd pr   « Reply #1 on: Jan 21st, 2017, 5:40pm » Quote Modify

The triplet (3, 5, 7) is the only set of "three consecutive odd numbers that are all prime" or, equivalently, the only triplet of prime numbers comprising an arithmetic progression with a common difference equal to 2.

Proof.

Let x be an odd prime > 3.

Consider the numbers x, x + 2 and x + 4.

Case 1). Let x = 3q + 1 where q = 2, 4, 6, ... Then the term x + 2 = 3q + 3 = 3(q + 1) is composite.

Case 2). Let x = 3q + 2 where q = 1, 3, 5, ... Then the term x + 4 = 3q + 6 = 3(q + 2) is composite. What was required to prove.

From where it follows that no other set of "three consecutive odd numbers that are all primes" exist. IP Logged
rmsgrey
Uberpuzzler           Gender: Posts: 2847 Re: sum of the squares of three consecutive odd pr   « Reply #2 on: Jan 22nd, 2017, 8:09am » Quote Modify

The stipulation that the primes be odd is not necessary - 4+9+25 is even so not a prime. IP Logged
towr
wu::riddles Moderator
Uberpuzzler      Some people are average, some are just mean.

Gender: Posts: 13730 Re: sum of the squares of three consecutive odd pr   « Reply #3 on: Jan 22nd, 2017, 12:01pm » Quote Modify

I think I'd interpret the problem as finding pn = pk2 + pk+12 + pk+22 where pi is the ith prime and i>1.
I mean, otherwise, why bring squaring and summing to a prime into it?

I don't see any immediate reason why there wouldn't be other solutions, but for primes below a billion it's the only solution.
 « Last Edit: Jan 22nd, 2017, 12:23pm by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
pex
Uberpuzzler      Gender: Posts: 880 Re: sum of the squares of three consecutive odd pr   « Reply #4 on: Jan 22nd, 2017, 1:23pm » Quote Modify

Proof: Any such triplet would only include primes greater than 3, which are well known to be equal to +/- 1 mod 6. That means their squares are all 1 mod 6, making the sum equal to 3 mod 6 and hence divisible by 3, so not prime. IP Logged

 Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »