$$2^{2^{2017}} + 3$$ is a prime number or not?

Note by Rony Phong
3 years, 5 months ago

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SR :)) I just fix it =))

- 3 years, 5 months ago

Nope. Apply mod 7 followed by Fermat's Little Theorem.

- 3 years, 5 months ago

By using Fermat's Little Theorem, we have : $2^{6} \equiv 1 \pmod{7} \Rightarrow 2^{8} = 2 ^{2^{3}} \equiv 4 \pmod{7}$

So that : $$2^{2^{4}} \equiv 8 \pmod{7}, 2^{2^{5}} \equiv 6 \pmod{7}, 2^{2^{6}} \equiv 2 \pmod{7}, 2^{2^{7}} \equiv 4 \pmod{7},....$$

We can see that if $$m = 4 \cdot n - 1$$ then $$2^{2^{m}} \equiv 4 \pmod{7}$$

Thus , $2^{2^{2017}} \equiv 6 \pmod{7}$ ... And I get stuck here :(

- 3 years, 5 months ago

- 3 years, 5 months ago

With tower of exponents, you should be applying Euler's Theorem. This tells us that we should find $$2^n \pmod{ \phi(7) }$$ first.

Staff - 3 years, 5 months ago

not a prime number

- 3 years, 5 months ago

I think it is prime because $$2^{2^{2017}}+1$$ is prime and many primes occur at difference of 2, so due to increased probability I chose it to be prime.

- 3 years, 5 months ago

I think $$2^{2^{2017}} + 1$$ is not a prime because it's is a Fermat number, and until now, there is just 5 Fermat number that are primes is : $$2^{2^{0}} + 1; 2^{2^{1}} + 1;2^{2^{2}} + 1;2^{2^{3}} + 1;2^{2^{4}} + 1$$

I have prove that $$2^{2^{2017}} + 3$$ is not a prime number, but thanks anyway.

- 3 years, 5 months ago

No.

Last few decimal digits of the solution are.... ...0172573696 <-Last Digit Even != Prime

Approx Solution:

4530133771486397128616413953101758728033979316264652841195935761051588450033937045213076515129395654998815159098974101615963200558154274787748738790211678644410538970061235159505665699096705221172128501364423224257417606736821962911252833066060363075687976750995313953457727475457383546376473713782226055256851957323635233060085750073840982862112794321584114078317329301232817710611834752430415334301714161673057478502612297701175639541114423640398305254720345616205423349569568002063030321637229012200090760969205506795214577344211530377676944078369371486135407432181252977569490258039805559010498267001268 decimal digits

~~ 4.530133771486397×10^606 decimal digits

- 3 years, 5 months ago

How are you finding the last decimal digit? Are you using a computer? If so, what is the extent of the rounding error?

It is obvious that this number is odd, and hence the last digit is not 6.

Staff - 3 years, 5 months ago