# How did fermat prove that a number can exist?

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Fermat used mathematical proof and the concept of modular arithmetic to show that certain numbers satisfy a specific equation.

## For those who need more details

Fermat’s proof that a number can exist was a groundbreaking contribution to the field of mathematics. He used his knowledge of modular arithmetic and mathematical proof to demonstrate the existence of numbers that satisfied specific equations.

Modular arithmetic allows for the manipulation of remainders when numbers are divided by a specific integer. Fermat used this concept to demonstrate that certain numbers, known as Fermat primes, exist. These numbers are of the form 2^(2^n) + 1, where n is a non-negative integer.

Fermat’s work in number theory went on to inspire generations of mathematicians. The famous mathematician Leonhard Euler even said of Fermat’s contributions, “Fermat’s theorems… are of the greatest elegance and set the mind of the mathematicians to work on the most beautiful problems.”

Interesting facts about Fermat and his contributions to mathematics include:

• Fermat was a lawyer by profession and did not pursue mathematics full-time.
• He famously wrote his mathematical discoveries in the margins of texts, leading to many of his results being lost or only partially preserved.
• Fermat’s Last Theorem, which states that x^n + y^n = z^n has no solutions for positive integers x, y, z, and n greater than 2, remained unproven for over 350 years.
• Andrew Wiles finally proved Fermat’s Last Theorem in 1994, using advanced mathematical concepts that were not available to Fermat during his lifetime.

As for a table, here are the first five Fermat primes:

n 2^(2^n) + 1
0 3
1 5
2 17
3 257
4 65537

## A visual response to the word “How did Fermat prove that a number can exist?”

This video presents a simplified version of Fermat’s Last Theorem with an added condition that n must be greater than or equal to z. The equation is rearranged and expressed as a product of two brackets with x and y being assumed to be less than z, leading to an inequality that can be solved using the previously added condition. The proof concludes with a contradiction that proves the initial statement that the equation has no non-trivial solutions.

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What that means in mathematical notation is that there are no integers x, y, z that satisfy the equation xⁿ + yⁿ = zⁿ when n is an integer greater than two. Fermat and other mathematicians were able to prove the conjecture for some specific values of n using the method of infinite descent also called Fermat’s method of descent.

The other answer is correct. In addition, there is significant evidence that Fermat did not have a proof of the theorem now known as Fermat’s Last Theorem.

First, we should note that Fermat was not a professional mathematician, only an amateur. He never published any mathematics himself. With just that, it would not seem strange that he did not publish his proof. However, his son Samuel decided to collect Fermat’s writings and letters. The famous margin which was too small for the proof was the margin of Fermat’s copy of Diophantus’s Arithmetica. Fermat likely wrote this note around 1630 when he first began studying this text.

From his writings and letters, we see a common trend. Almost all of the problems Fermat mentioned having solved were included in his work more than once, typically being restated as challenge problems which he then sent to various mathematicians with whom he was in correspondence. However, FLT appears only once, in this margin. It is never again mentioned in an…

## These topics will undoubtedly pique your attention

Herein, Did Fermat prove anything? Response to this: Fermat claimed to have found a proof of the theorem at an early stage in his career. Much later he spent time and effort proving the cases n=4 and n=5. Had he had a proof to his theorem earlier, there would have been no need for him to study specific cases.

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Subsequently, How do you prove Fermat theorem? Response: Let p be a prime and a any integer, then ap ≡ a (mod p). Proof. The result is trival (both sides are zero) if p divides a. If p does not divide a, then we need only multiply the congruence in Fermat’s Little Theorem by a to complete the proof.

What is Fermat last theorem for real numbers?
Fermat’s last theorem, also called Fermat’s great theorem, the statement that there are no natural numbers (1, 2, 3,…) x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2.

Also question is, What is the theory of Fermat?
As a response to this: Fermat’s theorem, also known as Fermat’s little theorem and Fermat’s primality test, in number theory, the statement, first given in 1640 by French mathematician Pierre de Fermat, that for any prime number p and any integer a such that p does not divide a (the pair are relatively prime), p divides exactly into ap − a.

Did Fermat have a proof?
Answer: Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat’s theorem on sums of two squares ), Fermat’s Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof.

Also Know, What is Fermat’s theorem?
The answer is: Fermat’s theorem, also known as Fermat’s little theorem and Fermat’s primality test, in number theory, the statement, first given in 1640 by French mathematician Pierre de Fermat, that for any prime number p and any integer a such that p does not divide a (the pair are relatively prime), p divides exactly into ap − a.

What is Wiles’s proof of Fermat’s Last Theorem? In reply to that: Wiles’s proof of Fermat’s Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet’s theorem, it provides a proof for Fermat’s Last Theorem.

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Simply so, Does Ribet prove Fermat’s Last Theorem? As an answer to this: Together with Ribet’s theorem, it provides a proof for Fermat’s Last Theorem. Both Fermat’s Last Theorem and the modularity theorem were almost universally considered inaccessible to prove by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge. : 203–205, 223, 226

Moreover, Did Fermat have a proof?
Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat’s theorem on sums of two squares ), Fermat’s Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof.

In this way, What is Fermat’s theorem? Fermat’s theorem, also known as Fermat’s little theorem and Fermat’s primality test, in number theory, the statement, first given in 1640 by French mathematician Pierre de Fermat, that for any prime number p and any integer a such that p does not divide a (the pair are relatively prime), p divides exactly into ap − a.

Similarly one may ask, What is Wiles’s proof of Fermat’s Last Theorem?
Answer: Wiles’s proof of Fermat’s Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet’s theorem, it provides a proof for Fermat’s Last Theorem.

Similarly one may ask, How many factors are there for Fermat numbers? The reply will be: Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, as of 2022, only to have been completely factored. The number of factors for Fermat numbers for , 1, 2,are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5,(OEIS A046052 ). Written out explicitly, the complete factorizations are (OEIS A050922 ).

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