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## 2^n+1 = np?

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• Hello! ... n=1, p=3; n=3, p=3. All other (2^n+1)/n are divisible by 3. Proof: Let p=(2^n+1)/n is prime greater than 3. n must be odd, so 2^n+1 is divisible by
Message 1 of 1 , May 30, 2001
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Hello!

Paul Mills wrote:

> So Q. 2^n + 1 = pn ? Yes! n=3 p=3.
> Interesting. So , can we have the next (n,p)
> please and lots more (infinite?).

n=1, p=3;

n=3, p=3.

All other (2^n+1)/n are divisible by 3.

Proof:

Let p=(2^n+1)/n is prime greater than 3. n must be odd, so
2^n+1 is divisible by 3; p is prime, hence, n is divisible
by 3.

So 2^n+1 is divisible by 2^3+1=9. Then n is divisible by 9
too, so 2^n+1 is divisible by 2^9+1=513=3^3*19, hence, n is
divisible by 3^3.

So 2^n+1 is divisible by 2^27+1=134217729=3^4*1657009,
hence, n is divisible by 3^4.

So 2^n+1 is divisible by 2^81+1...

Generally, 2^3^k+1 is divisible by 3^(k+1), hence, p is
always divisible by 3: contradiction.

We can obtain another interesting result:

Let n|(2^n+1). Then all prime factors of such n divide
2^3^k+1 for some k.

Proof is obvious from above.

Here's a complete list of such n's less than 10^8 and their
factorizations:

1 = 1
3 = 3
9 = 3^2
27 = 3^3
81 = 3^4
171 = 3^2*19
243 = 3^5
513 = 3^3*19
729 = 3^6
1539 = 3^4*19
2187 = 3^7
3249 = 3^2*19^2
4617 = 3^5*19
6561 = 3^8
9747 = 3^3*19^2
13203 = 3^4*163
13851 = 3^6*19
19683 = 3^9
29241 = 3^4*19^2
39609 = 3^5*163
41553 = 3^7*19
59049 = 3^10
61731 = 3^2*19^3
87723 = 3^5*19^2
97641 = 3^2*19*571
118827 = 3^6*163
124659 = 3^8*19
177147 = 3^11
185193 = 3^3*19^3
250857 = 3^4*19*163
263169 = 3^6*19^2
292923 = 3^3*19*571
354537 = 3^5*1459
356481 = 3^7*163
373977 = 3^9*19
531441 = 3^12
555579 = 3^4*19^3
752571 = 3^5*19*163
789507 = 3^7*19^2
878769 = 3^4*19*571
1063611 = 3^6*1459
1069443 = 3^8*163
1121931 = 3^10*19
1172889 = 3^2*19^4
1594323 = 3^13
1666737 = 3^5*19^3
1855179 = 3^2*19^2*571
2152089 = 3^4*163^2
2257713 = 3^6*19*163
2354697 = 3^3*87211
2368521 = 3^8*19^2
2636307 = 3^5*19*571
3190833 = 3^7*1459
3208329 = 3^9*163
3365793 = 3^11*19
3518667 = 3^3*19^4
4766283 = 3^4*19^2*163
4782969 = 3^14
5000211 = 3^6*19^3
5565537 = 3^3*19^2*571
6456267 = 3^5*163^2
6736203 = 3^5*19*1459
6773139 = 3^7*19*163
7064091 = 3^4*87211
7105563 = 3^9*19^2
7908921 = 3^6*19*571
9572499 = 3^8*1459
9624987 = 3^10*163
10097379 = 3^12*19
10556001 = 3^4*19^4
10970073 = 3^4*135433
14298849 = 3^5*19^2*163
14348907 = 3^15
15000633 = 3^7*19^3
16696611 = 3^4*19^2*571
19368801 = 3^6*163^2
20208609 = 3^6*19*1459
20319417 = 3^8*19*163
21192273 = 3^5*87211
21316689 = 3^10*19^2
22284891 = 3^2*19^5
23726763 = 3^7*19*571
28717497 = 3^9*1459
28874961 = 3^11*163
29884473 = 3^2*19*174763
30292137 = 3^13*19
31668003 = 3^5*19^4
32910219 = 3^5*135433
33894369 = 3^5*139483
35248401 = 3^2*19^3*571
38265939 = 3^7*17497
40889691 = 3^4*19*163^2
42896547 = 3^6*19^2*163
43046721 = 3^16
44739243 = 3^3*19*87211
45001899 = 3^8*19^3
50089833 = 3^5*19^2*571
55753011 = 3^2*19*571^2
57789531 = 3^5*163*1459
58106403 = 3^7*163^2
60625827 = 3^7*19*1459
60958251 = 3^9*19*163
63576819 = 3^6*87211
63950067 = 3^11*19^2
66854673 = 3^3*19^5
71180289 = 3^8*19*571
74214171 = 3^3*19*144667
86152491 = 3^10*1459
86624883 = 3^12*163
89653419 = 3^3*19*174763
90559377 = 3^4*19^3*163
90876411 = 3^14*19
95004009 = 3^6*19^4
98730657 = 3^6*135433
101683107 = 3^6*139483
105745203 = 3^3*19^3*571
114797817 = 3^8*17497
116226009 = 3^4*163*8803
122669073 = 3^5*19*163^2
127987857 = 3^5*19^2*1459
128689641 = 3^7*19^2*163
129140163 = 3^17
134217729 = 3^4*19*87211
135005697 = 3^9*19^3
143239347 = 3^4*19*163*571
150269499 = 3^6*19^2*571
167259033 = 3^3*19*571^2
173368593 = 3^6*163*1459
174319209 = 3^8*163^2
181877481 = 3^8*19*1459
182874753 = 3^10*19*163
190730457 = 3^7*87211
191850201 = 3^12*19^2
200564019 = 3^4*19^5
208431387 = 3^4*19*135433
213540867 = 3^9*19*571
222642513 = 3^4*19*144667
258457473 = 3^11*1459
259874649 = 3^13*163
268960257 = 3^4*19*174763
271678131 = 3^5*19^3*163
272629233 = 3^15*19
285012027 = 3^7*19^4
296191971 = 3^7*135433
305049321 = 3^7*139483
317235609 = 3^4*19^3*571
344380329 = 3^8*52489
344393451 = 3^9*17497
348678027 = 3^5*163*8803
350790507 = 3^4*163^3
368007219 = 3^6*19*163^2
383963571 = 3^6*19^2*1459
386068923 = 3^8*19^2*163
387420489 = 3^18
402653187 = 3^5*19*87211
405017091 = 3^10*19^3
423412929 = 3^2*19^6
429718041 = 3^5*19*163*571
450808497 = 3^7*19^2*571

Best wishes,

Andrey
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