I'm not sure which subforum this thread belongs in. It's a question with a mathematical background. The collection of the results is then done by a small program in assembly language,
written with the MASM64 SDK. There is nothing more to be said about this. I post it in the workshop. But if Hutch decides it should go somewhere else, that's okay too.
The two natural numbers m = 60 and n = 12 have a remarkable property. Their product is ten times greater than their sum. Therefore, the question is: Are there any other pairs of natural
numbers with this attribute? Yes, there are, but not many. The general solution is shown in the file sumprod.pdf. It's really not very complicated and explained in detail. The software shows
the 5 possible pairs of natural numbers and displays the results.
It'll look like this on the screen:
Number Pairs Sum Product
============ === =======
11 and 110 121 1210
12 and 60 72 720
14 and 35 49 490
15 and 30 45 450
20 and 20 40 400
Please, press any key to end the program...
If you're interested, you can take a look at the latex source I added to the archive. I hope the explanation and text are understandable.
I would be appreciated for any suggestions on how to improve it.
QuoteThe two natural numbers m = 60 and n = 10 have a remarkable property. Their product is ten times greater than their sum.
My grade school teachers must have shown me the wrong things.
According to them, their sum needs to be 60+10=70, and their product should be 60x10=600.
I fail to see that 600 is 10 times greater than 70, unless I misread some of the statement. :undecided:
(I agree the other example pairs match the description. :cool:)
It's probaby a typo, what surely Gunther meant was for m = 60 and n = 12.
Quote from: caballero on June 12, 2022, 03:31:53 AM
It's probaby a typo, what surely Gunther meant was for m = 60 and n = 12.
That's right. It should be n = 12. This is very unfortunate, because this error is also on p. 1 of the PDF file. I have corrected this and sumpror1.zip now contains the correct files.
Many thanks to Raymond and Caballero for their help. :thumbsup:
So is this some kind of remarkable property, or just a trivial coincidence? I don't know enough about number theory to have an opinion one way or the other.
I know there are all kinds of interesting, convoluted, strange things that happen with numbers and sequences of numbers ...
NoCforMe,
Quote from: NoCforMe on June 12, 2022, 05:23:10 AM
So is this some kind of remarkable property, or just a trivial coincidence?
It's a matter of taste or point of view - whatever you want. But is it really just a coincidence that applies to exactly 10 of an infinite number of natural numbers?
As already Euclid proved, there are infinitely many prime numbers. These have the remarkable property of being prime. What do you think would be the correct
description for these 10 natural numbers?
Quote from: Gunther on June 12, 2022, 05:58:39 AM
NoCforMe,
Quote from: NoCforMe on June 12, 2022, 05:23:10 AM
So is this some kind of remarkable property, or just a trivial coincidence?
It's a matter of taste or point of view - whatever you want. But is it really just a coincidence that applies to exactly 10 of an infinite number of natural numbers?
As already Euclid proved, there are infinitely many prime numbers. These have the remarkable property of being prime. What do you think would be the correct
description for these 10 natural numbers?
+INF float prime?
well if you test all primes that fit into a REAL10,its maybe more correct with infinity*percentage which is primes?maybe also is infinity
Hi,
Quote from: daydreamer on June 12, 2022, 11:21:13 PM
+INF float prime?
Not really. Primes are integers. While there are an infinite number of
real numbers, and a countable infinity of those are integer primes, that
seems a convoluted way of thinking about them.
Quotewell if you test all primes that fit into a REAL10,its maybe more correct with infinity*percentage which is primes?maybe also is infinity
No. REAL10 contains a finite number of bits. Therefore only a finite
number of numbers are possible. And since the exponent and sign bits
can be ignored, there are only 64 bits left to encode an integer. Definitely
only a finite number of primes can be represented in a REAL10. Or REAL*10
for us FORTRAN types.
Cheers,
Steve N.
Gunther,
Quote from: Gunther on June 11, 2022, 09:34:43 AM
... a remarkable property. Their product is ten times greater than their sum. Therefore, the question is: Are there any other pairs of natural
numbers with this attribute?
:thumbsup:
The property can be expanded to
m · n = q · (m + n) then
m = q + q ^ 2 / (n − q) and
m = n = 2 · q Interesting thing is that this hyperbolic function is close to more complex Lorentzian, that I use a lot fitting biological functions. I have to see if that can be replaced with this. :biggrin:
HSE
Daydreamer,
I think FORTRANS has already explained the essence of the matter. Here are some additional informations: The number M82589933 = 282589933 - 1 is probably the
largest known prime number. That's what we know so far. It has more than 24 million decimal digits. This would then also somewhat overwhelm the REAL10 type.
The number was found within the framework of GIMPS (https://www.mersenne.org/) (Great Internet Mersenne Prime Search). Mersenne prime numbers have the form Mn = 2n - 1. They are
named after Marin Mersenne (1588-1648), a French theologian and mathematician, and are especially well suited to search for large prime numbers.
The smallest Mersenne primes are: M2 = 22 - 1 = 3, M3 = 23 - 1 = 7 and M5 = 25 - 1 = 31. If M82589933 passes all subsequent tests, it would be the 51st Mersenne
prime number that was found.
HSE,
thank you for your interesting reply. :thumbsup:
Quote from: HSE on June 13, 2022, 02:26:51 AM
Interesting thing is that this hyperbolic function is close to more complex Lorentzian, that I use a lot fitting biological functions. I have to see if that can be replaced with this. :biggrin:
A biologist who knows mathematics. Excellent. Do you need such functions to solve Lotka-Volterra differential equations?
Gunther,
Quote from: Gunther on June 13, 2022, 02:51:36 AM
Do you need such functions to solve Lotka-Volterra differential equations?
No. But you remember me a classic: Noy-Meir. 1975. Stability of grazing systems: an application of predator-prey graphs. (I read that an eternity ago :biggrin:)
HSE
HSE,
Quote from: HSE on June 13, 2022, 05:15:40 AM
But you remember me a classic: Noy-Meir. 1975. Stability of grazing systems: an application of predator-prey graphs. (I read that an eternity ago :biggrin:)
This must be Imanuel Noy-Meir (https://plantscience.agri.huji.ac.il/imanuelnoymeir), right? An interesting guy.
Gunther,
Quote from: Gunther on June 13, 2022, 05:55:36 AM
This must be Imanuel Noy-Meir (https://plantscience.agri.huji.ac.il/imanuelnoymeir), right? An interesting guy.
Exactly. Just a couple of years before he died I saw him personally and I learned he born close here :biggrin:.
HSE.
HSE,
Quote from: HSE on June 13, 2022, 07:09:01 AM
Exactly. Just a couple of years before he died I saw him personally and I learned he born close here :biggrin:.
you were lucky to see him in a personal way with this. He was born in a hard time. That wasn't easy.
Gunther,
Quote from: Gunther on June 13, 2022, 08:22:04 AM
He was born in a hard time. That wasn't easy.
Yes. Here don't was the worst, but not easy at all. My father around that time play football in his town and all colonies around. He always telled us the fear of little boys because members of peronism (current governement) drived in the town with megafones saying "haga patria, mate un judío" (a lot of people now denie that). These little boys always asked shaking what happen to the few "criollos" they trust. It's presumed their fathers don't wanted to alarm them, and don't were saying to much.
Back in the subjec, I adapted the JJ number pairs code to ML64 (http://masm32.com/board/index.php?topic=10135.msg110609#msg110609). Very curious some
q have a lot of points and others only few or one!
HSE
HSE,
Quote from: HSE on June 13, 2022, 11:21:52 PM
Very curious some q have a lot of points and others only few or one!
Yes, indeed. But not so rare either. Please think only about the distribution of the prime numbers. They grow like wild weeds in the set of natural numbers. Even there
it's difficult to find the regularities.
Quote from: Gunther on June 14, 2022, 09:45:28 AM
HSE,
Quote from: HSE on June 13, 2022, 11:21:52 PM
Very curious some q have a lot of points and others only few or one!
Yes, indeed. But not so rare either. Please think only about the distribution of the prime numbers. They grow like wild weeds in the set of natural numbers. Even there
it's difficult to find the regularities.
but the composites have regularities
when I researched different ways to find Primes, I discovered the old multiplication table from school when I was kid,up to 120+ presented as GIF shows all composites have regularities
so all the gaps between for example between 6*2 and 7*2 = 13 is prime numbers
so I based a prime number checking algo,by MUL x*y loop saving into multiply LUT the composites,zeros left after zeroing it=primes
Daydreamer,
Quote from: daydreamer on June 14, 2022, 07:18:30 PM
but the composites have regularities
when I researched different ways to find Primes, I discovered the old multiplication table from school when I was kid,up to 120+ presented as GIF shows all composites have regularities
so all the gaps between for example between 6*2 and 7*2 = 13 is prime numbers
so I based a prime number checking algo,by MUL x*y loop saving into multiply LUT the composites,zeros left after zeroing it=primes
well, what can I say? There is one even prime number: the 2. All other prime numbers are odd. There is still an infinite number of them. Each of these prime numbers lies between two even numbers.
But how does that help us? The distance between the prime numbers is not regular - as far as we know. Perhaps the nontrivial zeros of the Riemann zeta function ζ (s) (https://en.wikipedia.org/wiki/Riemann_zeta_function) might help us with this
question, who knows? But so far, there has been no breakthrough in this direction.
Quote from: Gunther on June 15, 2022, 02:17:11 PM
Daydreamer,
Quote from: daydreamer on June 14, 2022, 07:18:30 PM
but the composites have regularities
when I researched different ways to find Primes, I discovered the old multiplication table from school when I was kid,up to 120+ presented as GIF shows all composites have regularities
so all the gaps between for example between 6*2 and 7*2 = 13 is prime numbers
so I based a prime number checking algo,by MUL x*y loop saving into multiply LUT the composites,zeros left after zeroing it=primes
well, what can I say? There is one even prime number: the 2. All other prime numbers are odd. There is still an infinite number of them. Each of these prime numbers lies between two even numbers.
But how does that help us? The distance between the prime numbers is not regular - as far as we know. Perhaps the nontrivial zeros of the Riemann zeta function ζ (s) (https://en.wikipedia.org/wiki/Riemann_zeta_function) might help us with this
question, who knows? But so far, there has been no breakthrough in this direction.
thanks,interesting link :thumbsup:
maybe in another dimension,or a dimension which doesnt have our usual time+3dimensions primes have a pattern?
look like caustics outside a boat
I dont remember if that was riemann_zeta,but there is a probability function of high primes returns probability for it might be a prime
I have SIMD experiment on test primes somewhere on forum,just to test use DIVPS and MULPS instead the usual div loop
whats your approach on prime testing?
the function 1/x in school I thought it might be related to wormhole shape
Gunther,
An additional feature is that maximum number always is less (but close) to (q + 1) ^2.
Eventually alway is q *(q+1), but that must be checked.
Updated in Re: 666, mathematics and Satan (http://masm32.com/board/index.php?topic=10135.msg110609#msg110609)
HSE
Daydreamer,
Quote from: daydreamer on June 15, 2022, 08:24:30 PM
thanks,interesting link :thumbsup:
thank you. Hopefully it helped you.
Quote from: daydreamer on June 15, 2022, 08:24:30 PM
maybe in another dimension,or a dimension which doesnt have our usual time+3dimensions primes have a pattern?
That's very speculative.
Quote from: daydreamer on June 15, 2022, 08:24:30 PM
I dont remember if that was riemann_zeta,but there is a probability function of high primes returns probability for it might be a prime
Do you mean the Prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem)? That' s very closely related to the zeta function.
Quote from: daydreamer on June 15, 2022, 08:24:30 PM
whats your approach on prime testing?
It depends on the upper limit. For small and medium values, I would use a sieve technique (Eratosthenes or Atkin). But at the moment this question doesn't appear for me,
because I am involved in certain astronomical calculations. Prime numbers don't play a role for that.
Quote from: daydreamer on June 15, 2022, 08:24:30 PM
the function 1/x in school I thought it might be related to wormhole shape
Wormholes. I've only seen them in certain movies: Star Trek, Deep Space Nine, Stargate SG-1, Déjà Vu, Contact, Donnie Darko etc. Do you know even one experimental result
for the existence of these theoretical objects?
HSE,
Quote from: HSE on June 16, 2022, 12:43:45 AM
An additional feature is that maximum number always is less (but close) to (q + 1) ^2.
Eventually alway is q *(q+1), ...
this is a remarkable phenomenon.
Quote from: HSE on June 16, 2022, 12:43:45 AM
... but that must be checked.
Checking isn't enough. It should be proofed. The question would be: Is that a worthwhile effort?