20070801, 18:04  #1 
Mar 2007
179 Posts 
Has this been proven? ... 10^n + 1
Has it been proven that 10^n + 1 is composite for all n > 2 ?
If so, can someone point me to it? It is composite for n odd, since 11 is a factor. But what about n even? 
20070801, 18:42  #2  
Feb 2005
11×23 Posts 
Quote:
Therefore, the only possibility for 10^n + 1 being prime is n = 2^k (similarly to Fermat primes). 

20070801, 19:11  #3  
"Bob Silverman"
Nov 2003
North of Boston
7464_{10} Posts 
Quote:
primes of the form 10^2^n + 1. 

20070801, 19:37  #4  
Mar 2007
179_{10} Posts 
Quote:
I believe my original post can be genralized a bit to... For any even base b, b^n + 1 is composite for all n odd, since b + 1 is a factor. My question is... What generalizations to your response can be made for even bases, b? Do all even bases behave "similar to Fermat primes" as you have proven for b = 10? 

20070801, 20:10  #5 
"William"
May 2003
New Haven
2^{2}·593 Posts 
Google "Generalized Fermat Numbers" to find more information.
I especially liked the list of factors in Table 1 of Bjorn and Reisel's 1998 paper Factors of Generalized Fermat Numbers 
20070801, 20:11  #6  
Feb 2005
375_{8} Posts 
Quote:
Quote:
If n has an odd factor m>1, then b^n + 1 has a nontrivial factor b^(n/m)+1. In particular, for odd n we can take m=n and obtain the property you mentioned. 

20070801, 20:14  #7  
∂^{2}ω=0
Sep 2002
República de California
10110111011100_{2} Posts 
Quote:
By way of example: 2^{2[sup]n}[/sup]+1 is prime for n=0,1,2,3,4 and likely for no other known values [and certainly not for n < 33]; 10^{2[sup]n}[/sup]+1 is prime for n=0,1 and for no other values n < 13. [As high as I tested using PARI just now  there is likely a larger known bound] 

20070802, 03:43  #8  
Feb 2006
Denmark
230_{10} Posts 
Quote:


20070802, 16:32  #9 
∂^{2}ω=0
Sep 2002
República de California
2^{2}×5×587 Posts 
"I am Hassan, the rebellious data point ... I scoff at your means and probability distribution functions. I and my fellow rebel outliers wage jihad against the 95%confidenceinterval infidel crusaders. The streets shall flow with the blood of the 3SD jackals and their lackeys! All praise be unto Allah, the Unstatistical."

20070803, 13:52  #10  
Feb 2007
1B0_{16} Posts 
Quote:
e.g. suppose we know that Fermat primes are finite, but the next and last one is 2072005925466 or larger than 2,365,100,000,000 (cf A090875) or 10^10^7 ? and/or, to what extend can heuristics which tell us that a sequence is "most probably" finite, also tell us something about the magnitude of the last term ? 

20070803, 14:40  #11  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×3×311 Posts 
Quote:
Your question strikes at the difference between existence proofs and constructive proofs. Or, (say) that knowing a diophantine equation has finitely many solutions, but not knowing a bound on them. Or knowing that a set is infinite but being unable to exhibit one of its elements or ....... any of a number of similar situations. Alan Baker's work on linear forms in logarithms can be useful, but it isn't always applicable. A striking example is the Catalan Conjecture. Before Preda Mihailescu finished his (very elegant) proof, we knew that there were at most finitely many solutions. And we had a bound. But the bound was beyond computer range. Then Preda found a connection to the Wieferich congruence and this brought the problem to a point where the computation became possible (but very large). Then he found a proof that avoided the computations. (APPLAUSE!) Another example: We know that almost all real numbers are Transcendental. The subset that is algebraic is countable, while the reals are uncountable. Thus, the algebraic numbers have density 0 in the reals. But knowing that almost all numbers are transcendental does not help us prove that e.g. The EulerMascheroni constant is. 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
List of proven/1k/2k/3k conjectures  The Carnivore  Conjectures 'R Us  84  20181206 09:34 
Primes for proven bases  CGKIII  Conjectures 'R Us  46  20170103 17:31 
Proven PRPs?  Random Poster  FactorDB  0  20120724 10:53 
Poincare conjecture proven?  Nebob  Lounge  36  20100330 13:14 
Are Legendre symbols proven to be defective?  jasong  Math  67  20080420 15:01 