Simple Question About Term(s) re: Fermat

[Janosh89]

Since Fermat, the French magistrate & noted mathematician, expounded :
all odd integers ,(2n+1) where n≥0, are representable by the difference of TWO squares
[1= 1[SUP]2[/SUP] -0² ]
so 3 = 2²-1²
5 = 3²-2²
7 = 4²-3²
and generally,
Z =y²-x²
and Z + x²= y² where y= Z/2 +1/2
[even integers of the form 4n are representable where y=Z/4+1 and x=y-2]

is there a term, in English language, to describe the x² mathematical term
that ,when added to Z, forms a perfect square, y²?
Thanks to anyone who will give me a definitive answer - I feel it should start with ad..

 

[jedishrfu]

I couldn't find a reference to the theorem you mentioned.

I did find Fermat's Sum of Two Squares theorem:

https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares

Is this what you meant? I don't see any special English mathematical name attached to the addends.

 

[Janosh89]

I think Fermat was following on from Gauss. Of course, Fermat's proof regarding the SUM of two squares, a²+ b² for positive integers of the form 12n+1 or 12n+5 ,gives us an equation for these
a²+ b²=y²- x²
I have omitted 12n+9 as they have multiple values, for each one, of y and x -at least one where
y=x+3

 

[mathman]

I think Fermat was following on from Gauss. Of course, Fermat's proof regarding the SUM of two squares, a²+ b² for positive integers of the form 12n+1 or 12n+5 ,gives us an equation for these
a²+ b²=y²- x²
I have omitted 12n+9 as they have multiple values, for each one, of y and x -at least one where
y=x+3

Fermat lived about 200 years before Gauss.

 

[Janosh89]

Thanks for putting me straight. My history of mathematical thought, like my diminishing memory capacity,
needs outside intervention.
Is the difference of two squares attributable to Pythagoras, Euler, ; not amongst the Elements, surely?
Please put me out of my misery. !

 

[StoneTemplePython]

I think all OP is looking for is

$\big(k+1\big)^2 - k^2 = \big(k^2 + 2k +1\big) -k^2 = \big(k^2 - k^2 \big) + 2k +1 = 2k +1$

It could be a named theorem, but I'm thinking its too simple to be named.

 

[Janosh89]

It obviously came into general mathematical thought very early on.
My point ,if you can call it that, about 12n+1, 12n+5 is that it directly relates to prime numbers; rather than the textbook usually given 4k+1.
I suppose I should say 1(MOD 4).
For 12n+5 integers , the only possible "target" value of y, for y², is y=3(MOD 6) where x is 2(MOD 6) or 4(MOD 6)
so for y=15
y²=225
29 = 225-14² Prime, since this is unique and no other target ,3(MOD 6) when squared can be achieved for any integer x
125= 225-10²
161= 225-8²
209= 225-4²
221= 225-2²
so this is obviously a first stage sieve, probably the only benefit it offering is that it eliminates
early on potentially large prime factors early on, close to the square root of Z, ≅ y