# Simple Question About Term(s) re: Fermat

• B
• Janosh89
In summary, Fermat's Sum of Two Squares theorem states that all odd integers, (2n+1) where n≥0, can be represented by the difference of two squares. This can be seen in the examples 3=22-12, 5=32-22, and 7=42-32. For even integers of the form 4n, they can be represented as Z=y2-x2, where y=Z/2+1/2. There is no specific English term to describe the mathematical term x2 that, when added to Z, forms a perfect square, y2. However, Ferm
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?

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.

StoneTemplePython
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.

jbriggs444
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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## 1. What is Fermat's Last Theorem?

Fermat's Last Theorem is a mathematical conjecture proposed by French mathematician Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.

## 2. Has Fermat's Last Theorem been proven?

Yes, Fermat's Last Theorem was finally proven by British mathematician Andrew Wiles in 1994, after more than 350 years of attempts by mathematicians around the world.

## 3. What is the significance of Fermat's Last Theorem?

Fermat's Last Theorem is considered one of the most famous and difficult theorems in mathematics. Its proof has had a major impact on the fields of number theory and algebraic geometry, and it has also inspired the development of new mathematical techniques.

## 4. Are there any other theorems named after Fermat?

Yes, there are several other theorems named after Fermat, including Fermat's Little Theorem, Fermat's Polygonal Number Theorem, and Fermat's Theorem on sums of two squares.

## 5. Who was Pierre de Fermat?

Pierre de Fermat (1601-1665) was a French lawyer and mathematician. He is best known for his contributions to number theory, including Fermat's Last Theorem, as well as his work in analytic geometry and the development of calculus.

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