A history of Greek mathematics - Wilbourhall.org

NUMBERS AS THE SUMS OF SQUARES 483
a given number is added to either part, the result will be a
square.' The condition is in two parts. There is no doubt as
to the first, 'The given number must not be odd' [i.e. no
number of the form 4n + 3 or 4 w, — 1 can be the sum of two
squares] ;
the text of the second part is corrupt, but the words
actually found in the text make it quite likely that corrections
made by Hankel and Tannery give the real meaning of the
original, ' nor must the double of the given number plus 1 be
measured by any prime number which is less by 1 than a
multiple of 4 '. This is tolerably near the true condition
stated by Fermat, The given number must not be odd, and
'
the double of it increased by 1 , when divided by the greatest
square which measures it, must not be divisible by a prime
number of the form 4 n — 1
.'
(ft)
On numbers which are the sum of three squares.
In V. 11 the number 3