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# 数学代写|数论代写Number Theory代考|Fermat Numbers

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## 数学代写|数论代写Number Theory代考|Fermat Numbers

In the last two sections we have been focused on the question: when is $2^n-1$ prime? A very similar question, of course, is: when is $2^n+1$ prime?
For an even number $2 n$, factoring $2^{2 n}-1$ involves factoring the two terms on the right-hand side of
$$2^{2 n}-1=\left(2^n-1\right)\left(2^n+1\right)$$
so it was quite natural for Fermat to become interested in the question of when $2^n+1$ is prime. Moreover, Fermat also realized that in order for $2^n+1$ to be prime, it is necessary for the exponent $n$ to be a power of 2 (see Problem 5.23).
This led him to make the following famous conjecture:
All numbers of the form $2^{2^n}+1$ are prime.
In 1640, he wrote to Frénicle with exactly this conjecture, listing the first seven such “primes,” that is, for $n=0,1,2,3,4,5,6$ :
$3,5,17,257,65537,4294967$ 297, 18446744073709551617 ,
saying
I don’t have an exact proof, but I have excluded such a great quantity of divisors by infallible demonstrations and I have shed so much light on the problem which will establish my idea, that it would be difficult for me to retract.

## 数学代写|数论代写Number Theory代考|Binomial Coefficients

Our final topic in this chapter on Fermat is binomial coefficients. We introduce this extremely useful subject at this point in order to be able to discuss how Fermat was able to use binomial coefficients to prove a theorem about which he wrote, “there can hardly be found a more beautiful or a more general theorem about numbers.” That seems like a theorem that might be worth talking about. But first, we need to talk about binomial coefficients.

Diagrams that contained the binomial coefficients-that is, the coefficients for binomials such as $(a+b)^2,(a+b)^3, \ldots$-appeared in China as early as the eleventh century. An elegant diagram showing these coefficients arranged in a triangle for all these binomials up through $(a+$ $b)^8$ appeared with the title page of Zhu Shijie’s book The Precious Mirror of the Four Elements in 1303. Another of China’s great mathematicians, Yang Hui, had included a similar triangular array in his own series of works which appeared from 1261 to 1275 .

It is easy to expand a binomial such as $(a+b)^3$ using Zhu Shijie’s diagram because the third row of the diagram gives us the coefficients $1,3,3,1$ for this binomial, so we know that
$$(a+b)^3=1 \cdot a^3+3 \cdot a^2 b+3 \cdot a b^2+1 \cdot b^3 .$$
Note that in this diagram we count the rows starting from the top with 0 rather than 1 , so that the third row will correspond to a binomial of degree 3.

We can even read the numbers in the sixth row without much difficulty, and conclude that
$$(a+b)^6=1 \cdot a^6+6 \cdot a^5 b+15 \cdot a^4 b^2+20 \cdot a^3 b^3+15 \cdot a^2 b^4+6 \cdot a b^5+1 \cdot b^6 .$$

## 数学代写|数论代写Number Theory代考|Fermat Numbers

$$2^{2 n}-1=\left(2^n-1\right)\left(2^n+1\right)$$

1640年，他写信给fracimnicle，提出了这个猜想，列出了$n=0,1,2,3,4,5,6$的前七个“质数”:
$3,5,17,257,65537,4294967$ 297, 18446744073709551617，

## MATLAB代写

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