How Can a Number Be Written as a Sum of Squares or Primes?

[abiyo]

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

 

[CRGreathouse]

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

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

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.

 

[abiyo]

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)