Hell number formulas

[sweety439]

A hell number formula is defined as a formula containing only small numbers and the only variable n is the exponent, and the formula only produce nonprime numbers for all small positive integers n, but not for all n (i.e. it does not have covering congruence, algebraic factorization, or combine of them)

e.g.

n*2^n+1 (for n > 1)
n*2^n+1 for prime n
5^n+8 (for n > 1)
(18^n-1)/17 (for n > 2)
4*72^n-1 (72 = 2^3*3^2 is the smallest Achilles number)
4*30^n-1 (for n > 3) (30 = 2*3*5 is the product of the first 3 primes)
8*48^n-1 (for n > 1) (48 = 2^4*3)
3*432^n-1 (432 = 2^4*3^3)
8*432^n-1 (432 = 2^4*3^3)
36*48^n+1
2^n-n-2 (for n > 3)
(2^n-6)*2^n+1 (for n > 3)
(2^n-7)*2^n+1
12^n+1 (for n > 1) (this is special case, with Fermat number behavior, only power-of-2 n can give primes)
Smarandache numbers in base 4

Non-examples:

4*9^n-1 (with algebraic factorization)
4*24^n-1 (with combine of coveted congruence and algebraic factorization)
(2^n-5)*2^n+1 (with covering congruence)
(2^n-2)*2^n+1 (with algebraic factorization)

[MattcAnderson]

Interesting observation Sweety439.

Have a nice day.
Matt

[Uncwilly]

Originally Posted by sweety439

A hell number formulas is defined

by who? You or the broader community. If the latter, please provide a link.

[sweety439]

Originally Posted by Uncwilly

by who? You or the broader community. If the latter, please provide a link.

by me.

[science_man_88]

Originally Posted by sweety439

by me.

Then it should read "Let a hell number be defined as..."

[gd_barnes]

You realize that many of these forms have primes, don't you?

[sweety439]

Originally Posted by gd_barnes

You realize that many of these forms have primes, don't you?

But none of them have small primes, e.g. the most famous 4*72^n-1 (4 is the smallest composite number and 72 is the smallest Achilles number), it has no prime with n >= 1 until n = 1119849

[gd_barnes]

Originally Posted by sweety439

But none of them have small primes, e.g. the most famous 4*72^n-1 (4 is the smallest composite number and 72 is the smallest Achilles number), it has no prime with n >= 1 until n = 1119849

Define small. Some of these have primes for n<10000. I'd call those small compared to what we typically search.

There are an infinite number of forms like this with no "small" primes. What is the point that you are trying to make?

[sweety439]

Originally Posted by gd_barnes

Define small. Some of these have primes for n<10000. I'd call those small compared to what we typically search.

There are an infinite number of forms like this with no "small" primes. What is the point that you are trying to make?

Forms that only contain 5-smooth (at least 7-smooth) numbers

For the case n*2^n+1 with n>1, since it only contains the number 2, the first prime is at n=141 is unusual large, the same for 5^n+8 with n>1 whose first prime is n=95 (it corresponds to the largest minimal prime in base 5, i.e. 5^95+8, or 1(0^93)13 in base 5, see below) since it only contains 5 and 8

Also, I have a project of minimal elements for the prime strings > b written in base b, for bases 2<=b<=36, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are completely solved, bases 11, 22, 30 are also completely solved if strong probable primes > 10^25000 are allowed, for the data of minimal primes > b and the unsolved families, see https://github.com/xayahrainie4793/quasi-mepn-data, the unsolved family 9{5} (i.e. 95, 955, 9555, 95555, …) in base 13 contain no primes with length < 116000, but cannot be proven to only contain composites by covering congruence, algebraic factorization, or combine of them

[xilman]

Originally Posted by sweety439

Forms that only contain 5-smooth (at least 7-smooth) numbers

For the case n*2^n+1 with n>1, since it only contains the number 2, the first prime is at n=141 is unusual large, the same for 5^n+8 with n>1 whose first prime is n=95 (it corresponds to the largest minimal prime in base 5, i.e. 5^95+8, or 1(0^93)13 in base 5, see below) since it only contains 5 and 8

Also, I have a project of minimal elements for the prime strings > b written in base b, for bases 2<=b<=36, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are completely solved, bases 11, 22, 30 are also completely solved if strong probable primes > 10^25000 are allowed, for the data of minimal primes > b and the unsolved families, see https://github.com/xayahrainie4793/quasi-mepn-data, the unsolved family 9{5} (i.e. 95, 955, 9555, 95555, …) in base 13 contain no primes with length < 116000, but cannot be proven to only contain composites by covering congruence, algebraic factorization, or combine of them

My offer to prove some of those primes still stands.