A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

PORISMS AND PROPOSITIONS ASSUMED 481
In fact YZ±a = {(m+l)(2m+l) + 2a} 2 ,
ZX±a = {m(2m+ l) + 2a} 2 ,
XY±a= {.m(m+l) + a} 2 .
(3) If
X = m 2 + 2, F=(m+l)*+2, Z = 2{m 2 + (m + 1) 2 + 1 ]
+ 2,
then the six expressions
F^_(F+if), #X-(#+X), XY-(X+Y)
}
YZ-X, ZX-Y, XY-Z
are all squares (V. 6).
In fact
YZ- (Y + Z) = {2m 1 + 3m + 3)
2
, F^-I=(2m
2 + 3m + 4) 2 ;
&c.
2. The second class is much more important, consisting of
propositions in the Theory of Numbers which we find first
stated or assumed in the Arithmetica. It was in explanation
or extension of these that Fermat's most famous notes
were written. How far Diophantus possessed scientific proofs
of the theorems which he assumes must remain largely a
matter of speculation.
(a)
Theorems on the composition of numbers as the sum
of two squares.
(1) Any square number can be resolved into two squares in
any number of ways (II. 8).
(2) Any number which is the sum of two squares can be
resolved into two other squares in any number of ways (II. 9).
(It is implied throughout that the squares may be fractional
as well as integral.)
(3) If there are two whole numbers each of which is the
sum of two squares, the product of the numbers can be
resolved into the sum of two squares in two ways.
In fact (a 2 + b 2 ) (c2 + d 2 )
= (ac ± bd) 2 + {ad + be) 2 .
This proposition is used in III. 19, where the problem is
to find four rational right-angled triangles with the same
1523.2 I i
482 DIOPHANTUS OF ALEXANDRIA
hypotenuse. The method is this. Form two right-angled
triangles from (a, b) and (c, d) respectively, by which Diophantus
means, form the right-angled triangles
(a 2 + b 2 , a 2 -b 2 , 2ab) and (c 2 + d 2 , c 2 -d 2 , 2cd).
Multiply all the sides in each triangle by the hypotenuse of
the other; we have then two rational right-angled triangles
with the same hypotenuse (a 2 + b 2 ) (c 2 + d 2 ).
Two others are furnished by the formula above; for we
have only to form two right-angled triangles ' ' from (ac + bd,
ad — be) and from (ac — bd, ad + be) respectively. The method
fails if certain relations hold between a, b, c, d. They must
not be such that one number of either pair vanishes, i.e. such
that ad = be or ac = bd, or such that the numbers in either
pair are equal to one another, for then the triangles are
illusory.
In the case taken by Diophantus a 2 + b 2 = 2 2 + 2 = 5,
c 2 + d 2 = 3 2 + 2 2 = 1 3, and the four right-angled triangles are
(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33).
On this proposition Fermat has a long and interesting note
as to the number of ways in which a prime number of the
form 4 n + 1 and its powers can be (a) the hypotenuse of
a rational right-angled triangle, (b) the sum of two squares.
He also extends theorem (3) above : If a prime number which
'
is the sum of two squares be multiplied by another prime
number which is also the sum of two squares, the product
will be the sum of two squares in two ways ; if the first prime
be multiplied by the square of the second, the product will be
the sum of two squares in three ways ; the product of the first
and the cube of the second will be the sum of two squares
in four ways, and so on ad infinitum'
Although the hypotenuses selected by Diophantus, 5 and 13,
are prime numbers of the form 4 n + 1 , it is unlikely that he
was aware that prime numbers of the form 4 n + 1 and
numbers arising from the multiplication of such numbers are
the only classes of numbers which are always the sum of two
squares ;
this was first proved by Euler.
(4) More remarkable is a condition of possibility of solution
prefixed to V. 9, 'To divide 1 into two parts such that, if