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
257 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
7475_{10} Posts 
Quote:
primes of the form 10^2^n + 1. 

20070801, 19:37  #4  
Mar 2007
179 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
4505_{8} 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
257 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
11,743 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
2·5·23 Posts 
Quote:


20070802, 16:32  #9 
∂^{2}ω=0
Sep 2002
República de California
11743_{10} 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
2^{4}·3^{3} 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
1D33_{16} 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. 

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