I can prove the following facts:

* There are precisely 2438 numbers which cannot be written as the sum of
distinct squared primes (n.b. **Pebody** does not take "1" as prime)

* There are infinitely many numbers which cannot be
written as the sum of
at most 15 distinct squared primes

(Indeed the numbers 840k+795) are of this form

* All numbers smaller than 100000 can be written as the sum of at most 16
distinct squared primes. Thus **conjecture**:

**L=17164, K=16, N={1,2,3,5,6,...,16251,16322,17163}, M={}, S=17.**

Claim: Every number n>=20000 can be written as the sum of distinct
primes.

Theorem 1: For real n>25, there exists a prime between n and 1.2n
exclusive (Nagura 1952)

Theorem 2: Every number 17164<=n<=100000 can be written as the sum of
distinct primes. (By checking)

Corollary 3: Every number 17164<=n can be written as the sum of distinct
primes.

Proof:

Let N be the smallest number above 20000 that cannot be written as the
sum of distinct primes. Note by Theorem 2, that N>100000.

Let n be Sqrt(N/2) in theorem 1. Then there exists a prime p such that
Sqrt(N/2)<p<1.2*Sqrt(N/2).

Therefore

N/2<p^2<0.72N

Now N/2<p^2 means that N<2p^2, which means that N-p^2<p^2. Therefore,
since N cannot be written as the sum of distinct square primes (including
p^2), N-p^2 cannot be written as the sum of square primes.

However N-p^2<N. It must follow that N-p^2<2000. However,
N-p^2>0.28N>28000. QED

There are 2438 distinct numbers which cannot be written as such starting
1,2,3,5,6,7,8,10,11,12,14,... and ending 14570, 14595, 14642, 15002, 15243,
15314, 15435, 15842, 16251, 16322, 17163.

All numbers other than these, up to 100000 can be written as the sum of
at most 16 distinct squared primes. The first number taking 16 distinct
squared primes is 15075.

Theorem

If a number of the form 120k+75 is the sum of fewer than 16 distinct
squared primes, it is the sum of exactly 3 distinct squared primes, one of
which is 25.

Proof:

Looking (mod 8):

each squared prime is either 4 (once) or 1 (all other times).

Therefore to get to 3 (mod 8), you must either use 4 and 7(mod 8) others,
or not use 4 and use 3(mod 8) in total.

Thus you must use 3,8 or 11 or at least 16.

Looking (mod 3)

each squared prime is either 0 (once) or 1 (all other times)

Therefore to get to 0(mod 3) you must either use
0,1,3,4,6,7,9,10,12,13,15 or at least 16.

Putting these together you must use 3 or at least 16.

Looking (mod 5)

each squared prime is either 0 (once) or 1,4 (all other times)

therefore to get to 0 (mod 5) using 3, you must use 0 (once) and 1,4. QED

Corollary:

There are infinitely many numbers that are not the sum of fewer than 16
distinct squared primes.

Proof:

Every number of the form 840k+795=120(7k+6)+75 is not the sum of fewer
than 16 distinct squared primes. If it were, then it would be the sum of
precisely 3 distinct squared primes, one of them being 25.

Therefore 840k+770 would be the sum of 2 distinct squared primes.
However,

7|x^2+y^2 => 7|x and 7|y. Therefore both of these distinct primes

are divisible by 7.