1105 as Sum of Two Squares
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Theorem
$1105$ can be expressed as the sum of two squares in more ways than any smaller integer:
\(\ds 1105\) | \(=\) | \(\, \ds 1089 + 16 \, \) | \(\, \ds = \, \) | \(\ds 33^2 + 4^2\) | ||||||||||
\(\ds \) | \(=\) | \(\, \ds 1024 + 81 \, \) | \(\, \ds = \, \) | \(\ds 32^2 + 9^2\) | ||||||||||
\(\ds \) | \(=\) | \(\, \ds 961 + 144 \, \) | \(\, \ds = \, \) | \(\ds 31^2 + 12^2\) | ||||||||||
\(\ds \) | \(=\) | \(\, \ds 625 + 529 \, \) | \(\, \ds = \, \) | \(\ds 24^2 + 23^2\) |
Proof
Here is the source code of a program in Python that finds all positive integers up to $1105$ that can be written as a sum of two squares in more ways than any smaller positive integer:
import numpy as np def two_sq_decomp_rich(n): bound = int(np.floor(np.sqrt(n))) count_of_two_sq_decomps = [] for i in range(2*(bound + 1)*(bound + 1)): count_of_two_sq_decomps.append(0) for i in range(bound+1): for j in range(i+1): count_of_two_sq_decomps[i*i+j*j] = count_of_two_sq_decomps[i*i+j*j] + 1 max_sq_decomps = 0 sq_decomp_rich_numbers = [] for i in range(n+1): if count_of_two_sq_decomps[i] > max_sq_decomps: max_sq_decomps = count_of_two_sq_decomps[i] sq_decomp_rich_numbers.append(i) return sq_decomp_rich_numbers print(two_sq_decomp_rich(1105))
Output:
[0, 25, 325, 1105]
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1105$
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1105$