B Simple Question About Term(s) re: Fermat

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The discussion revolves around Fermat's theorem on the representation of odd integers as the difference of two squares, specifically the mathematical expression Z = y² - x². Participants seek a specific term in English to describe the x² term that, when added to Z, results in a perfect square y². While references to Fermat's Sum of Two Squares theorem and its historical context with Gauss are made, no definitive term for the addends is identified. The conversation also touches on the relationship between certain integer forms and prime numbers, suggesting a connection to modular arithmetic. Overall, the thread highlights the complexities and historical nuances of mathematical concepts related to Fermat's work.
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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..
 
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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
 
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Janosh89 said:
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.
 
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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. !
 
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.
 
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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
 
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