Sum of Squares, Distinct Primes

by abiyo
Tags: distinct, primes, squares
 P: 43 Hi All, So I was just wondering if there is a formula for the number of ways a number can be written as a sum of squares?(Without including negatives, zero or repeats). For example 5=2^2+1^2. (There is only one way for 5). Second question along this line is: In how many ways can a number be written as a sum of primes(i.e a sum of two primes, three primes ). Third Question: 10=2+3+5 Thus 10 can be written as a sum of maximum three prime numbers; no more. Is there such an upper bound for other numbers? I was doing this for small numbers but would be interesting to see if there is some sort of pattern or theory Thanks a lot Abiyo
HW Helper
P: 3,684
 Quote by abiyo So I was just wondering if there is a formula for the number of ways a number can be written as a sum of squares?(Without including negatives, zero or repeats). For example 5=2^2+1^2. (There is only one way for 5).
This is complicated, see
http://mathworld.wolfram.com/SumofSquaresFunction.html

 Quote by abiyo Second question along this line is: In how many ways can a number be written as a sum of primes(i.e a sum of two primes, three primes ).
About exp(2 Pi sqrt(n/log n) / sqrt(3)). I don't imagine there is a nice closed-form formula.
http://oeis.org/A000607

 Quote by abiyo Third Question: 10=2+3+5 Thus 10 can be written as a sum of maximum three prime numbers; no more. Is there such an upper bound for other numbers? I was doing this for small numbers but would be interesting to see if there is some sort of pattern or theory
Can you be more specific? This is ambiguous.
 P: 43 Thanks CRGreatHouse. Sorry the last question is worded badly. What I wanted to ask is Pick an integer n. We want to find partition of n into its prime parts. For example 10=7+3 10=2+3+5 There are two partitions of 10 into primes. The first one involves two primes, the second involves three primes. The claim then is that 3 is the maximum partition of 10 into primes. 3 is the longest partition. Now let me choose some arbitrary integer(large n). I might have x number of partitions of n into prime parts. I want to determine the longest partition. (how many prime numbers are involved at maximum). Is there a formula or a theoretical treatment? Thanks a lot once again (My English is terrible. sorry if this is confusing again)

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