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Will Jagy
Will Jagy
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EDIT, Sunday afternoon. Every time I raise the multiple of square bound, I get new sums of two squares, which makes the previous count incorrect. I am going to just post these as they print out: the norms are correct, each coefficient is doubled, so coefficients are either all odd or all even. Divide by two to get the actual coefficients.

 13 :    -1    1   -5   -5       729 :   -45   21   15   15       784 :    44  -20  -20  -20   
4.63905
      norm    two squares    not        total 
      13         252          84         336

EDIT, Sunday afternoon. Every time I raise the multiple of square bound, I get new sums of two squares, which makes the previous count incorrect. I am going to just post these as they print out: the norms are correct, each coefficient is doubled, so coefficients are either all odd or all even. Divide by two to get the actual coefficients.

 13 :    -1    1   -5   -5       729 :   -45   21   15   15       784 :    44  -20  -20  -20   
4.63905
      norm    two squares    not        total 
      13         252          84         336
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Will Jagy
Will Jagy
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  • 121

EDIT, Sunday: I ran norm 121 overnight out of curiousity, with bound $6 n^2,$ which may or may not have really been large enough to correctly count the two squares. An extreme was $q = r^2 + s^2$ with $$ q = \frac{ 7}{2 } + \frac{ 11 }{2 } i + \frac{ 1 }{2 } j - \frac{ 17 }{2 } k, \; $$ $$ r^2= \frac{ 407}{2 } + \frac{ 155 }{2 } i + \frac{ 341 }{2 } j + \frac{ 31 }{2 } k, \; \; \; s^2 = -200 -72 i -168 j -24 k $$ with norms $121,76729,73984,$ and $76729 / 14641 \approx 5.241$

EDIT, Sunday: I ran norm 121 overnight out of curiousity, with bound $6 n^2,$ which may or may not have really been large enough to correctly count the two squares. An extreme was $q = r^2 + s^2$ with $$ q = \frac{ 7}{2 } + \frac{ 11 }{2 } i + \frac{ 1 }{2 } j - \frac{ 17 }{2 } k, \; $$ $$ r^2= \frac{ 407}{2 } + \frac{ 155 }{2 } i + \frac{ 341 }{2 } j + \frac{ 31 }{2 } k, \; \; \; s^2 = -200 -72 i -168 j -24 k $$ with norms $121,76729,73984,$ and $76729 / 14641 \approx 5.241$

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Will Jagy
Will Jagy
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I wrote a program; I like to know what I'm getting into before trying to prove things. I can already suggest that it appears all Hurwitz quaternions are expressible as $$ q = u (r^2 + s^2), $$ where $q,u,r,s$ are Hurwitz quaternions and $u$ is a unit (there are 24 of those). For any fixed norm, so far at least one out of six such quaternions are the sum of two squares, and multiplication on the left by a unit gives everything. The worst behavior is norm 21, only 136 out of 768 Hurwitz quaternions of norm 21 are the sum of two squares.

About the sizes of things, in finding the sums of two squares (before multiplying by any unit): so far it has sufficed to take $r,s$ with norms less than double the norm of $q,$ so the norms of their squares are less that 4 times the square of the norm of $q.$ The extremes for this seem to occur when the norm of $q$ is already the square of a prime $p \equiv 3 \pmod 4.$ So, two examples are $q = r^2 + s^2$ with $$ q = \frac{ 5}{2 } - \frac{ 3 }{2 } i - \frac{ 1 }{2 } j - \frac{ 1 }{2 } k, \; \; r^2= \frac{ 19}{2 } + \frac{ 21 }{2 } i + \frac{ 7 }{2 } j + \frac{ 7 }{2 } k, \; \; s^2 = -7 - 12 i - 4 j - 4 k $$ with norms $9,225,225,$ and $225 / 81 \approx 2.777$

Next $q = r^2 + s^2$ with $$ q = \frac{ 13}{2 } - \frac{ 5 }{2 } i - \frac{ 1 }{2 } j - \frac{ 1 }{2 } k, \; $$ $$ r^2= -\frac{ 61}{2 } - \frac{ 165 }{2 } i - \frac{ 33 }{2 } j - \frac{ 33 }{2 } k, \; s^2 = 37 +80 i +16 j +16 k $$ with norms $49,8281,8281,$ and $8281 / 2401 \approx 3.449$

If need be I can find out what the squares are squares of. Part of a big speed improvement was dropping that printout.

Alright, sssssatistics. As I said, before multiplying on the left by the 24 units, the sums of two squares are not usually all items of that norm. For example, in norm $1,$ the six Hurwitz quaternions $\pm i, \pm j, \pm k$ are not the sum of two squares. Not my fault.

  norm     two squares   not         total 
   1          18           6          24
   2           6          18          24
   3          68          28          96
   4          24           0          24
   5          84          60         144
   6          24          72          96
   7         144          48         192
   8          18           6          24
   9         162         150         312
  10          42         102         144
  11         180         108         288
  12          88           8          96
  13         228         108         336
  14          48         144         192
  15         432         144         576
  16          24           0          24
  17         180         252         432
  18          84         228         312
  19         392          88         480
  20         120          24         144
  21         136         632         768   
  22         120         168         288
  23         432         144         576
  24          96           0          96
  25         390         354         744
  26          90         246         336
  27         724         236         960
  28         184           8         192
  29         276         444         720
  30         192         384         576
  31         600         168         768
  32          24           0          24
  33         564         588        1152
  34         114         318         432
  35         864         288        1152
  36         240          72         312
  37         564         348         912
  38         168         312         480
  39         848         496        1344
  40         138           6         144
  41         588         420        1008
  42         192         576         768  
  43         792         264        1056  
  44         264          24         288    
  45         900         972        1872      
  46         216         360         576     
  47         936         216        1152    
  48          88           8          96    
  49         642         726        1368

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