20120504, 12:54  #1 
Feb 2011
Singapore
5×7 Posts 
Binomial Primes
[P.S. Sorry if i posted in the wrong section, and sorry for my bad English too]
Please note that prime numbers here excludes 1 and includes 2. Firstly, allow me to introduce and define some terms. Binomial numbers : Numbers of the form a^n +/ b^n, where a,b,n are integers >1 Decremental binomial numbers : Binomial numbers of the form (a+1)^n a^n Decremental binomial primes : Decremental binomial numbers which are prime Binomial exponent : The value of n in a decremental binomial number of the form (a+1)^n a^n, where a,n are integers >1 Binomial base : The value of a in a decremental binomial number of the form (a+1)^n a^n, where a,n are integers >1 I have came out with the following statements, and the tried to prove/disprove them. Statement 1 : All primes of the form a^n  b^n, where a,b,n are natural numbers, must be decremental primes. Statement 2 : All decremental binomial primes must have a prime binomial exponent. Statement 3 : All decremental binomial primes must have a prime binomial base. Statement 4 : There exists infinitely many decremental binomial primes. Statement 5 : There exists a deterministic test to determine the primality of a decremental binomial number for all positive binomial exponents and binomial base. So far, I have only managed to prove statements 1 and 2, and disprove statement 3. (Although i am unsure of the validity of my proofs) Proof of statement 1 : Let a^n  b^n be a prime, where a,b,n are integers >1. By polynomial factorization, a^n  b^n = (a  b)[a^(n1) + a^(n2)b +... + ab^(n2) + b^(n1)] It follows that a  b is a factor of a^n  b^n. Since a^n  b^n is a prime, a  b = +/(a^n  b^n) or +/ 1 The only solution to a  b = +/(a^n  b^n) is a = b. For a = b, a^n  b^n = 0, hence a  b = +/1 For a  b = 1, a = b  1. For a = b  1, a^n  b^n <= 0. Therefore, a = b + 1. It follows that for a binomial number of the form a^n  b^n to be prime, it must be a decremental binomial number. QED Proof of statement 2 : Let (a+1)^n  a^n be a prime, where a,n are natural numbers. Suppose n is composite. It can then be expressed in the form n = r X s for some r and s. By binomial expansion, (a+1)^n  a^n = (a+1)^(rs)  a^(rs) =[(a+1)^r  a^r][a^(r(s1)) + a^(r(s2))b +... + b^(s(r1))] Hence, if n is composite, (a+1)^n  a^n will always be factorized into 2 distinct numbers, making it composite, contradicting it being prime. It follows that the binomial exponent must be prime. QED (Dis)Proof of statement 3: A counter example is 7^2  6^2 = 13 Here, the binomial base is 6, which is composite. However, i have yet to proof or disprove statements 4 and 5, and i am here today to kindly ask you for your help in helping me with improving the statements and their respective proofs/disproofs, and also to proof/disproof statements 4 and 5. If possible, it would be great to come out with a deterministic test for decremental binomial numbers and a proof of correctness for this test as well. Last fiddled with by fivemack on 20120504 at 14:19 
20120504, 13:07  #2  
"Forget I exist"
Jul 2009
Dartmouth NS
2×3×23×61 Posts 
Quote:


20120504, 13:13  #3 
Feb 2011
Singapore
23_{16} Posts 

20120504, 13:16  #4  
"Forget I exist"
Jul 2009
Dartmouth NS
2·3·23·61 Posts 
Quote:
Quote:
Last fiddled with by science_man_88 on 20120504 at 13:17 

20120504, 13:18  #5 
Feb 2011
Singapore
23_{16} Posts 
Sorry sir, i missed out a decremental before binomial prime in statement 1. Fixed it now.

20120504, 13:26  #6 
"Forget I exist"
Jul 2009
Dartmouth NS
2·3·23·61 Posts 

20120504, 13:32  #7  
Feb 2011
Singapore
5·7 Posts 
Quote:
Thank you for pointing out, ive now limited n to integers more than one in the definition of a binomial number. Last fiddled with by Lee Yiyuan on 20120504 at 13:33 

20120504, 13:41  #8  
"Forget I exist"
Jul 2009
Dartmouth NS
8418_{10} Posts 
Quote:
Quote:
of course: OEIS seq A138389 : Quote:
Last fiddled with by science_man_88 on 20120504 at 13:44 

20120504, 13:48  #9  
Feb 2011
Singapore
5×7 Posts 
Quote:
Fixed once again. I am indeed in debt for your guidiance. 

20120504, 13:50  #10 
"Forget I exist"
Jul 2009
Dartmouth NS
8418_{10} Posts 

20120504, 14:00  #11  
Feb 2011
Singapore
5·7 Posts 
Quote:
On a related note, have you any idea how to contact the admins? I seem to have troubles editing my thread post as i need to change "natural numbers" to "integers more than 1" 

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