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# Tag Archives: Leonhard Euler

## Euler’s (Prime to) Prime Generating Equation

The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent. The prime generating equation by Euler is a binomial which is actually very specific and yields more primes than any other relations out there in number theory. Euler told that the equation $f(x)=x^2+x+k$ yields many prime numbers with the values of x being input from x=0 to x=k-2; k being a prime.

Let’s see how many primes we can get by using different values of k and x:

 Serial Number Value of k (prime) Value of x (from x=0 to x=k-2) Value of f(x)=(x^2+x)+k Not a Prime? 1 2 0 2 2 3 0 3 3 1 5 4 5 0 5 5 1 7 6 2 11 7 3 17 8 7 0 7 9 1 9 No 10 2 13 11 3 19 12 4 27 No 13 5 37 14 11 0 11 15 1 13 16 2 17 17 3 23 18 4 31 19 5 41 20 6 53 21 7 67 22 8 83 23 9 101 24 13 0 13 25 1 15 No 26 2 19 27 3 25 No 28 4 33 No 29 5 43 30 6 55 No 31 7 69 No 32 8 85 NO 33 9 103 34 10 123 No 35 11 145 No 36 17 0 17 37 1 19 38 2 23 39 3 29 40 4 37 41 5 47 42 6 59 43 7 73 44 8 89 45 9 107 46 10 127 47 11 149 48 12 173 49 13 199 50 14 227 51 15 257 52 19 0 19 53 1 21 54 2 25 No 55 3 31 56 4 39 No 57 5 49 58 6 61 59 7 75 No 60 8 91 No 61 9 109 62 10 129 No 63 11 151 64 12 175 No 65 13 201 No 66 14 229 67 15 259 68 16 291 69 17 325 No 70 23 0 23 71 1 25 No 72 2 29 73 3 35 No 74 4 43 75 5 53 76 6 65 No 77 7 79 78 8 95 No 79 9 113 80 10. 133 81 11 155 No 82 12 179 83 13 205 No 84 14 233 85 15 263 86 87 16 295 No 88 17 329 89 18 365 No 90 19 403 91 20 443 92 21 485 No

The above table yields many prime numbers, which again can be put at the place of k and so on the table can be progressed.

According to Euler, 41 was the most appropriate value of k yielding more prime numbers than any other k. In the list below, each value of f(x) is a prime for k=41:

 k 41 x 0 f(x) 41 1 43 2 47 3 53 4 61 5 71 6 83 7 97 8 113 9 131 10 151 11 173 12 197 13 223 14 251 15 281 16 313 17 347 18 383 19 421 20 461 21 503 22 547 23 593 24 641 25 691 26 743 27 797 28 853 29 911 30 971 31 1033 32 1097 33 1163 34 1231 35 1301 36 1373 37 1447 38 1523 39 1601

So, the Euler’s Prime Generating Equation can be written as
$f(x) = x^2+x+41$ ; where x is an integer ranging from 0 to 39.

Wait. What if we increase the value of x beyond the limit of 39? What will we get?

The next values of f(x) in this series would be 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393, … .
Are all these prime numbers too? The answer is no. 1681 is not a prime number, neither are 1763 and 2021. Though all others are prime numbers.

## Fermat Numbers

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$.

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$ etc.

Fermat observed that all the integers $F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $F_n$ is a prime for each natural value of $n$.

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $F_1 -F_4$ are primes but $F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

# Theorem:

The Fermat number $F_5$ is divisible by $641$ i.e., $641|F_5$.

# Proof:

As defined $F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$

Factorising $641$ in such a way that $641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $a=5 \bigwedge b=2^7$ we have $ab+1=641$.

Subtracting $a^4=5^4=625$ from 641, we get $ab+1-a^4=641-625=16=2^4 \ \ldots (2)$.

Now again, equation (1) could be written as
$F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $641|F_n$.

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $F_4$. The best guess is that all Fermat numbers $F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $F_n, F_m \ m > n \ge 0$, $\mathrm{gcd}(F_m, F_n) =1$.

Following two theorems are very useful in determining the primality of Fermat numbers:

# Pepin Test:

For $n \ge 1$, the Fermat number $F_n$ is prime $\iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$

# Euler- Lucas Theorem

Any prime divisor $p$ of $F_n$, where $n \ge 2$, is of form $p=k \cdot 2^{n+2}+1$.

Fermat numbers ($F_n$) with $n=0, 1, 2, 3, 4$ are prime; with $n=5,6,7,8,9,10,11$ have completely been factored; with $n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $n=14,20,22,24$ have no factors known but proved composites. $F_{33}$ has not yet been proved either prime or composite.

## Solving Ramanujan’s Puzzling Problem

Consider a sequence of functions as follows:-

$f_1 (x) = \sqrt {1+\sqrt {x} }$
$f_2 (x) = \sqrt{1+ \sqrt {1+2\sqrt {x} } }$

$f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } }$

……and so on to

$f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } }$

Evaluate this function as n tends to infinity.

Or logically:

Find

$\displaystyle{\lim_{n \to \infty}} f_n (x)$ .