1105 as Sum of Two Squares

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]

Try it online!

$\blacksquare$

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): $1105$