Help me please!

222017+3 2^{2^{2017}} + 3 is a prime number or not?

Note by Rony Phong
4 years, 3 months ago

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SR :)) I just fix it =))

Rony Phong - 4 years, 3 months ago

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

Pi Han Goh - 4 years, 3 months ago

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By using Fermat's Little Theorem, we have : 261(mod7)28=2234(mod7)2^{6} \equiv 1 \pmod{7} \Rightarrow 2^{8} = 2 ^{2^{3}} \equiv 4 \pmod{7}

So that : 2248(mod7),2256(mod7),2262(mod7),2274(mod7),....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=4n1m = 4 \cdot n - 1 then 22m4(mod7)2^{2^{m}} \equiv 4 \pmod{7}

Thus , 2220176(mod7)2^{2^{2017}} \equiv 6 \pmod{7} ... And I get stuck here :(

Rony Phong - 4 years, 3 months ago

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@Rony Phong Check your working again. Your entire 3rd line is wrong.

Pi Han Goh - 4 years, 3 months ago

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

Calvin Lin Staff - 4 years, 3 months ago

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not a prime number

Ramiel To-ong - 4 years, 3 months ago

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I think it is prime because 222017+12^{2^{2017}}+1 is prime and many primes occur at difference of 2, so due to increased probability I chose it to be prime.

Department 8 - 4 years, 3 months ago

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I think 222017+12^{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 : 220+1;221+1;222+1;223+1;224+12^{2^{0}} + 1; 2^{2^{1}} + 1;2^{2^{2}} + 1;2^{2^{3}} + 1;2^{2^{4}} + 1

I have prove that 222017+32^{2^{2017}} + 3 is not a prime number, but thanks anyway.

Rony Phong - 4 years, 3 months ago

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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

Jack Bennett - 4 years, 3 months ago

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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.

Calvin Lin Staff - 4 years, 3 months ago

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