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Sum of Squares, Distinct Primes

  1. Nov 12, 2010 #1
    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
     
  2. jcsd
  3. Nov 12, 2010 #2

    CRGreathouse

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    This is complicated, see
    http://mathworld.wolfram.com/SumofSquaresFunction.html

    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

    Can you be more specific? This is ambiguous.
     
  4. Nov 12, 2010 #3
    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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