Number pairs with a remarkable property.

[Gunther]

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:

Code Select Expand


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.

[raymond]

The 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.

(I agree the other example pairs match the description. )

[avcaballero]

It's probaby a typo, what surely Gunther meant was for m = 60 and n = 12.

[Gunther]

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. 

[NoCforMe]

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 ...

[Gunther]

NoCforMe,

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?

[daydreamer]

NoCforMe,

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

[FORTRANS]

Hi,

+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.

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

   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.

[HSE]

Gunther,

... 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?

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. 

HSE

[Gunther]

Daydreamer,

I think FORTRANS has already explained the essence of the matter. Here are some additional informations: The number M_82589933 = 2^82589933 - 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 (Great Internet Mersenne Prime Search). Mersenne prime numbers have the form M_n = 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 M_82589933 passes all subsequent tests, it would be the 51st Mersenne
prime number that was found.

[Gunther]

HSE,

thank you for your interesting reply. 

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. 

A biologist who knows mathematics. Excellent. Do you need such functions to solve Lotka-Volterra differential equations?

[HSE]

Gunther,

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  )

HSE

[Gunther]

HSE,

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  )

This must be Imanuel Noy-Meir, right? An interesting guy.

[HSE]

Gunther,

This must be Imanuel Noy-Meir, right? An interesting guy.

Exactly. Just a couple of years before he died I saw him personally and I learned he born close here .

HSE.

[Gunther]

HSE,

Exactly. Just a couple of years before he died I saw him personally and I learned he born close here .

you were lucky to see him in a personal way with this. He was born in a hard time. That wasn't easy.