Binomial Primes

[Lee Yiyuan]

[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^(n-1) + a^(n-2)b +... + ab^(n-2) + b^(n-1)]
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(s-1)) + a^(r(s-2))b +... + b^(s(r-1))]
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.

[science_man_88]

Originally Posted by Lee Yiyuan

[P.S. Sorry if i posted in the wrong section, and sorry for my bad English too]

Firstly, allow me to introduce and define some terms.

Binomial numbers : Numbers of the form a^n +/- b^n

Decremental binomial numbers : Binomial numbers of the form (a+1)^n -a^n

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

Binomial base : The value of a in a decremental binomial number of the form (a+1)^n -a^n


I have came out with the following statements, and the tried to prove/disprove them.

Statement 1 : For a binomial number to be prime, it must be a decremental binomial number.

Statement 2 : For a decremental binomial number to be prime, it's binomial exponent must be prime.

Statement 3 : For a decremental binomial number to be prime, it's binomial base must be prime.

Statement 4 : There exists infinitely many binomial primes.

Statement 5 : There exists a deterministic test to determine the primality of a decremental binomial number.


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.
By binomial factorization,
a^n - b^n = (a - b)[a^(n-1) + a^(n-2)b +... + ab^(n-2) + b^(n-1)]
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 to be prime, it must be of the form (a+1)^n - a^n. QED

Proof of statement 2 :
Let (a+1)^n - a^n be a prime.
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^(rs) - a^(rs)
=[(a+1)^r - a^r][a^(r(s-1)) + a^(r(s-2))b +... + b^(s(r-1))]
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.

a=2; 3^2+2^2 -> 9+4->13 a prime

[Lee Yiyuan]

Originally Posted by science_man_88

a=2; 3^2+2^2 -> 9+4->13 a prime

True, but what i meant by binomial primes were numbers of the form (a+1)^n - a^n
This of course is my own definition of a binomial prime (i couldnt find any definitions after a google search on the term)

[science_man_88]

Originally Posted by Lee Yiyuan

True, but what i meant by binomial primes were numbers of the form (a+1)^n - a^n
This of course is my own definition of a binomial prime (i couldnt find any definitions after a google search on the term)

For a binomial number to be prime, it must be a decremental binomial number.

you define "a binomial number" as a^nb^n so it's possible to have 3^2+2^2 as a binomial number and it's prime and it's not a decremental binomial number under your definitions. this equality disproves your first statement.

[Lee Yiyuan]

Originally Posted by science_man_88

you define "a binomial number" as a^nb^n so it's possible to have 3^2+2^2 as a binomial number and it's prime and it's not a decremental binomial number under your definitions. this equality disproves your first statement.

Sorry sir, i missed out a decremental before binomial prime in statement 1. Fixed it now.

[science_man_88]

Originally Posted by Lee Yiyuan

Sorry sir, i missed out a decremental before binomial prime in statement 1. Fixed it now.

ran a PARI/gp script this is also false if you have no limit on n :

10^1-3^1 -> 7 -> prime 10-3 =7 not 1.

[Lee Yiyuan]

Originally Posted by science_man_88

ran a PARI/gp script this is also false if you have no limit on n :

10^1-3^1 -> 7 -> prime 10-3 =7 not 1.

Thank you for pointing out, ive now limited n to integers more than one in the definition of a binomial number.

[science_man_88]

Binomial primes : Decremental binomial numbers which are prime

Statement 1 : For a binomial number of the form a^n - b^n to be prime, it must be a decremental binomial number.

these 2 clearly show the same thing, namely, that you only intend that decremental binomials can be prime.

of course:

OEIS seq A138389 :

Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

though I feel like I forgot to leave criticism for R.D.Silverman to make.

[Lee Yiyuan]

Originally Posted by science_man_88

these 2 clearly show the same thing, namely, that you only intend that decremental binomials can be prime.

of course:

OEIS seq A138389 :

though I feel like I forgot to leave criticism for R.D.Silverman to make.

Fixed once again. I am indeed in debt for your guidiance.

[science_man_88]

Originally Posted by Lee Yiyuan

Fixed once again. I am indeed in debt for your guidiance.

sad part is I'm one that needs the guidance half the time.

[Lee Yiyuan]

Originally Posted by science_man_88

sad part is I'm one that needs the guidance half the time.

But you have determination, and i respect you for that!
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"