Are All Imaginable Integer Series Necessarily Infinite?

[thetexan]

TL;DR

Conjecture

Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size.

Examples…

Prime numbers
Mersenne primes
Odd perfect numbers(if they exist)
Zeroes of the Zeta function

Regardless of how crazy such as…

The series of numbers whose prime factors are all Mersenne and that have a perfect odd number (if they exist) immediately following it.

The conjecture states that if there are such numbers, there will be infinitely many of them.

In other words, if a series of integers can be imagined there will necessarily be an infinite number of them.

If this conjecture is true, then all other conjectures which ponder the size of such sets are moot.

Anyway, that’s my conjecture

Tex

 

[PeroK]

You couldn't have researched this very thoroughly. I knew two off the top of my head.

1) The number of faces on a Platonic solid:

https://en.wikipedia.org/wiki/Platonic_solid

2) There are finitely many sporadic simple groups:

https://en.wikipedia.org/wiki/Sporadic_group

Plus the trivial example of even prime numbers.

I found this one, which is the sequence of numbers that are superabundant and highly composite:

https://oeis.org/A166981

 

[martinbn]

In addition to @PeroK answer, your example of the zeta function doesn't fit. The zeros are not a list of increasing integer numbers.

 

[Vanadium 50]

TL;DR Summary: Conjecture

Mersenne primes

Is this set known to be infinite?

 

[PeroK]

Is this set known to be infinite?

I think the Texan's conjecture is that if there is one, there is an infinite number. And that would resolve the open question on Mersenne primes, among many other open questions.

 

[FactChecker]

TL;DR Summary: Conjecture

Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size.

Anyway, that’s my conjecture

You might have it backwards. Conjectures that deal with a finite number of things to consider may be easier to prove or disprove, therefore removing them from the "famous conjecture" category. The remaining ones are often a conjecture that there are an infinite number, so whether they are finite or infinite IS the conjecture itself.

 

[Vanadium 50]

So this thread is about conjectures of conjectures?
My brain hurts.

 

[gmax137]

There are infinitely many false conjectures. That's my conjecture. It may be false.

 

[WWGD]

" The list of even primes larger than 2"?
" The Real zeros of $x^2+a, a \in \mathbb R^{+}$?"

 

[docnet]

"The number of cookies Bob has left, if he began the recess with 300 cookies and ate 299. "?

 

[WWGD]

There are infinitely many false conjectures. That's my conjecture. It may be false.

Self-reference. Godel Alert!

 

[thetexan]

I think that’s my point. If indeed, by this conjecture, there must always be an infinite number of members of a set of any series which is based or dependent on, or draws from the infinite set of numbers, then trying to prove a series is infinite is pointless. We can know that it must be.

For example. Take Grahams number. If we want to know how many multiples of G64 are products of only 2 Mersenne primes….

Such as G64^2, G64^3…etc. How many of those numbers are the products of only 2 Mersenne primes. You can try to prove that there are infinitely many.

Or, if my conjecture is proven, then I know the answer to the above and it’s already proven.

Like Godel’s incompleteness theorem. It sorta takes the motivation out of trying to prove anything.

Tex

 

[PeroK]

I think that’s my point. If indeed, by this conjecture, there must always be an infinite number of members of a set of any series which is based or dependent on, or draws from the infinite set of numbers, then trying to prove a series is infinite is pointless. We can know that it must be.

So, your conjecture is that any sequence of positive integers is infinite? Note that a series is a sum.

So, for example, there must be an infinite sequence of even prime numbers? There can't, by your conjecture, only be one even prime number.

 

[thetexan]

But, my conjecture is this…if someone DID find an even prime, there would have to be an infinite number of them.

Tex

 

[WWGD]

But, my conjecture is this…if someone DID find an even prime, there would have to be an infinite number of them.

Tex

I found one. It's $2$. Where is/are any other(s)? Or at least a proof that there are?

 

[Vanadium 50]

But, my conjecture is this…if someone DID find an even prime, there would have to be an infinite number of them.

Is this supposed to be a serious thread?

 

[Drakkith]

Thread locked for moderation.

 

[berkeman]

Thread will remain closed. From the Mentor discussion about this thread closure:

It's either wrong or trivial.
"numbers below 6" is a counterexample to the conjecture.
Unless we require every element in the set to have a larger successor, but then it's trivial that it is infinite and OP didn't add anything.