Sum of Squares, Distinct Primes

In summary, 3 is the maximum number of ways a number can be written as a sum of squares. There is only one way for 5, and there are two ways for 10 to be written as a sum of primes.
  • #1
abiyo
43
0
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
 
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  • #2
abiyo said:
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

abiyo said:
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

abiyo said:
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.
 
  • #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)
 

What is the sum of squares of distinct primes?

The sum of squares of distinct primes refers to the sum of the squares of all the prime numbers that are used to make up a given number. For example, the number 30 can be written as 2^1 x 3^1 x 5^1, so its sum of squares of distinct primes would be (2^2 + 3^2 + 5^2) = 38.

What is the formula for calculating the sum of squares of distinct primes?

The formula for calculating the sum of squares of distinct primes is n = p1^2 + p2^2 + p3^2 + ... + pn^2, where n is the given number and p1, p2, p3, etc. are its distinct prime factors.

What is the significance of sum of squares of distinct primes in mathematics?

The sum of squares of distinct primes is significant in mathematics because it helps in solving problems related to prime numbers, factorization, and divisibility. It is also used in areas such as number theory, cryptography, and data encryption.

Can the sum of squares of distinct primes be negative?

No, the sum of squares of distinct primes cannot be negative as it involves adding the squares of numbers, which are always positive. However, the result of this calculation can be negative if the given number itself is negative.

Are there any patterns or relationships between the sum of squares of distinct primes and the given number?

Yes, there are certain patterns and relationships between the sum of squares of distinct primes and the given number. For example, a number is a perfect square if and only if its sum of squares of distinct primes is equal to the number itself. Additionally, there are various rules and theorems that can be used to determine the sum of squares of distinct primes for different types of numbers.

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