all prime numbers are equivalent in value?

MattcAnderson · 2021-11-20, 01:52

The prime pages are so cool.
Chris Caldwell. Has helped me.
See. Primes.utm.edu

MattcAnderson · 2022-02-15, 02:23

To put it simply, all integers have the same value.

A larger number is not better than a smaller one.
This also goes for prime numbers.
Just because I do calculations of small numbers, doesn't mean that those data are less important than giant Mersenne numbers.

That is all.

paulunderwood · 2022-02-15, 05:12

Quote:

Originally Posted by MattcAnderson

To put it simply, all integers have the same value.

A larger number is not better than a smaller one.
This also goes for prime numbers.
Just because I do calculations of small numbers, doesn't mean that those data are less important than giant Mersenne numbers.

That is all.

This is subjective. Would you want your bank transactions or your private communications dependent on "small primes"? (You now need more than 500,000 digits to make the top5000 primes -- there are 1,276 mega-primes).

VBCurtis · 2022-02-15, 05:30

If you think Mersenne primes unimportant, I suggest you find a forum not named after them to post your ramblings.

axn · 2022-02-15, 08:40

One way to define the value of something is to look at its rarity. In that sense, bigger primes are rarer, and therefore more valuable than smaller ones.

Another way is to look at how costly it is to "produce" something. In that sense, primes that took longer to compute are more valuable than primes that are easier to compute.

Yet another way is to look at the utility of something. In that sense bigger prime numbers are more valuable, say, in cryptographic application. Of course, the mersenne primes aren't useful for these, because they are too rare.
OTOH, mersenne primes (and mersenne factors and fully-factored mersennes) have their own mathematical applications as well.

Finally, in the subjective sense, your own mathematical musings are more important to you than other people's. Only in this very narrow sense that your original statement is accurate. So please do what you want. However, if you're posting about it in public, take care to present the results in a meaningful easy-to-understand way. Keep things well-organized.

Dr Sardonicus · 2022-02-15, 13:18

You have not explained what you mean by "value." From what you have written, your meaning seems to be "I like."

This seems to me to be a narrow view. The integers are a mental construct of the human mind, a pure abstraction, and have no intrinsic value. Any value to humanity that can be assigned to them is extrinsic. An individual likes pairs of small primes which differ by, say, 46. OK, fine. But just about anyone with a computer and an Internet connection can reproduce a list of such primes on their own in seconds, if they also happen to like them.

OTOH, dealing with Mersenne numbers having even modestly large prime exponents requires more computing effort than any one individual can command. And so it came to pass, that GIMPS was born. The importance of large Mersenne numbers has gone far beyond what Martin Gardner called "the useless elegance of perfect numbers" or whether Mersenne's guesses of which prime exponents yielded prime numbers were correct.

Indeed, were it not for the pursuit of finding factors of Mersenne numbers and identifying Mersenne primes, this Forum would not exist, so you would not have the opportunity to post here. Thus, "giant Mersenne numbers" have an importance to you personally which pairs of small primes differing by 46 do not.

Some possibly interesting questions about pairs of primes differing by 46: They need not be consecutive primes. In fact, consecutive primes differing by 46 are fairly thin on the ground, at least initially, compared to all pairs of primes differing by 46. so, one may ask: How many "admissible k-tuples" are there whose first and last terms differ by 46? How large can k be? And what is the relative contribution of each to the pairs of primes differing by 46?

Dr Sardonicus · 2022-02-15, 13:48

It occurred to me that, for each prime p, there is a "non-Archimedean valuation" (p-adic valuation) of the integers/rational numbers, and there is a p-adic completion for each. However, the valuations defined by two different primes are inequivalent. (This means that the p-adic valuations for different primes define different topologies.)

Uncwilly · 2022-02-15, 15:16

Quote:

Originally Posted by axn

Yet another way is to look at the utility of something.

Are not then 2, 3, 5, 7, 11, 13, 17, 19, 23 very valuable in sieving. The list of the first 100 primes are quite useful for doing the first test of primality on other numbers.

retina · 2022-02-15, 15:21

Quote:

Originally Posted by MattcAnderson

... all integers have the same value.

What about the poor forgotten real numbers? Are they of lesser value?

And ℚ? ℂ?

MattcAnderson · 2022-02-15, 15:26

First, I appreciate all the input. I respect your opinions. Now I copy and paste from note pad and add another file.

I did not say that Mersenne primes are
unimportant.

I simply stated that small primes are not
unimportant.

My blog posts are not ramblings. That is your
opinion.

As far as I know, nobody has shared a list of
2 tuples 46 apart, before I did it. This data
needs to be shared.

I admit that some of my calculations are quick,
like one second of computer time. I have done
other prime constellation calculations that are
more that 2 hours of computer calculation
(or more that 2 days!)

This data is rare because if you want to see it,
you have to calculate it yourself. It is not
yet shared.

I love the Mersenne project. Don't get me wrong.
I have a computer running 24/7 toward our cause.
I also like to do my own coding sometimes.

Also, this 'blog is public so I appreciate other
people's opinions.

Another good point. There is no immediate
monitary value in an integer. It is just a
number. It may take weeks of computer time
to be sure about a number, or it may take a
short time to churn out a new data set.

Both are important.

I like that phrase "the useless elegance of
perfect numbers". So far, less that 60
perfect numbers are known, but if we, as a team,
keep using our computers as calculators, we will
find more. Maybee one a year or less, but we
are making progress. It is worth it.

Another good point, when the pairs of prime
numbers are 46 apart, or more, then they are not
neccessarily consecutive prime numbers.

Many people know that for prime numbers bigger
than 6, all prime numbers are of the form
1/-1 mod 6
and when we consider mod 30, the wheel becomes
more complicated.

I am not making an apology, and I am not done
with my "pairs of prime numbers" project.

There is more to do on this project.

how large can 'k' be for k-tuples? This is a good question.

My intuition is k can be arbitrarily large.

There is no 'conspiracy' in the number line, to
quote Terrance Tao.

Have a nice day.

2021-11-20, 01:52	#1
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 11·109 Posts	all prime numbers are equivalent in value? The prime pages are so cool. Chris Caldwell. Has helped me. See. Primes.utm.edu

2022-02-15, 02:23	#2
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 10010101111₂ Posts	all prime numbers are equivalent in value? To put it simply, all integers have the same value. A larger number is not better than a smaller one. This also goes for prime numbers. Just because I do calculations of small numbers, doesn't mean that those data are less important than giant Mersenne numbers. That is all.

2022-02-15, 13:48	#7
Dr Sardonicus Feb 2017 Nowhere 13×17×29 Posts	all prime numbers are equivalent in value? NOT! It occurred to me that, for each prime p, there is a "non-Archimedean valuation" (p-adic valuation) of the integers/rational numbers, and there is a p-adic completion for each. However, the valuations defined by two different primes are inequivalent. (This means that the p-adic valuations for different primes define different topologies.)

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2022-02-15, 05:30	#4
VBCurtis "Curtis" Feb 2005 Riverside, CA 2×2,927 Posts	If you think Mersenne primes unimportant, I suggest you find a forum not named after them to post your ramblings.

2022-02-15, 08:40	#5
axn Jun 2003 2³×683 Posts	One way to define the value of something is to look at its rarity. In that sense, bigger primes are rarer, and therefore more valuable than smaller ones. Another way is to look at how costly it is to "produce" something. In that sense, primes that took longer to compute are more valuable than primes that are easier to compute. Yet another way is to look at the utility of something. In that sense bigger prime numbers are more valuable, say, in cryptographic application. Of course, the mersenne primes aren't useful for these, because they are too rare. OTOH, mersenne primes (and mersenne factors and fully-factored mersennes) have their own mathematical applications as well. Finally, in the subjective sense, your own mathematical musings are more important to you than other people's. Only in this very narrow sense that your original statement is accurate. So please do what you want. However, if you're posting about it in public, take care to present the results in a meaningful easy-to-understand way. Keep things well-organized.

2022-02-15, 13:18	#6
Dr Sardonicus Feb 2017 Nowhere 13·17·29 Posts	You have not explained what you mean by "value." From what you have written, your meaning seems to be "I like." This seems to me to be a narrow view. The integers are a mental construct of the human mind, a pure abstraction, and have no intrinsic value. Any value to humanity that can be assigned to them is extrinsic. An individual likes pairs of small primes which differ by, say, 46. OK, fine. But just about anyone with a computer and an Internet connection can reproduce a list of such primes on their own in seconds, if they also happen to like them. OTOH, dealing with Mersenne numbers having even modestly large prime exponents requires more computing effort than any one individual can command. And so it came to pass, that GIMPS was born. The importance of large Mersenne numbers has gone far beyond what Martin Gardner called "the useless elegance of perfect numbers" or whether Mersenne's guesses of which prime exponents yielded prime numbers were correct. Indeed, were it not for the pursuit of finding factors of Mersenne numbers and identifying Mersenne primes, this Forum would not exist, so you would not have the opportunity to post here. Thus, "giant Mersenne numbers" have an importance to you personally which pairs of small primes differing by 46 do not. Some possibly interesting questions about pairs of primes differing by 46: They need not be consecutive primes. In fact, consecutive primes differing by 46 are fairly thin on the ground, at least initially, compared to all pairs of primes differing by 46. so, one may ask: How many "admissible k-tuples" are there whose first and last terms differ by 46? How large can k be? And what is the relative contribution of each to the pairs of primes differing by 46?