[QUOTE=mfgoode]I give two numbers each distinctive in their own ways and mathematically connected to each other. This is a good one for you fetofs as its original.
137
3425
Mally :coffee:[/QUOTE]

since a week with no response I'll post soemthing close, but not quite:
137 is the smallest three-digit number such that all two-digit combinations of distinct digits are primes
3425 = 25*137 is such that all three-digit combinations of distinct digits are composite except for 523

let me know if this is even close to the correct mindset
-Tom :help:

[QUOTE=tom11784]137 is the smallest three-digit number such that all two-digit combinations of distinct digits are primes[/QUOTE]

hmmm, try 113

Special whole numbers
 

[QUOTE=tom11784]since a week with no response I'll post soemthing close, but not quite:
137 is the smallest three-digit number such that all two-digit combinations of distinct digits are primes
3425 = 25*137 is such that all three-digit combinations of distinct digits are composite except for 523

let me know if this is even close to the correct mindset
-Tom :help:[/QUOTE]
:smile:
Numbers: "hmmm, try 113". Pretty good buddy!
Tom 11784: Good observation on 3425.

See what you can make of this.
The reciprocal of this number 137 figured early in the theory of Relativity and Quantum Electrodynamics. This dimensional constant (DC) was obtained from the formula 2**e^2hc = 1/137 known as the constant of fine structure.
Later it was replaced by the constant (DC)15.
The number 137 has unique mathematical properties.
Multiply 137 by any natural number. Take the number block formed by the last two digits of the result. Square it and add to the square of the number block formed by the rest of the result. The sum is always divisible by 137.

Eg : 137*14 = 19 18
19^2 + 18^2 = 685 = 137*5

The number 137 has properties linked with the number 34 25

34^2 + 25^2 = 137*13 = 17 81
17^2 + 81^2 = 137*50 = 68 50
68^2 + 50^2 = 137*52 = 71 24
71^2 + 24^2 = 137*41 = 56 17
56^2 + 17^2 = 137*25 = 34 25

1) Take a number block of 5 digits
137*91 = 124 67 (5 digits)
124^2 + 67^2 = 198 65 = 137*45

2) Take a number block of 6 digits
137*911 = 1248 07 (6 digits)
1248^2 + 07^2 = 15575 53 = 137*11369

and so on…

Amazing isn’t it? :bounce:
Mally :coffee:

[QUOTE=Numbers]hmmm, try 113[/QUOTE]
oops - smallest with distinct digits such that ...

[quote=mfgoode]The number 137 has unique mathematical properties.
Multiply 137 by any natural number. Take the number block formed by the last two digits of the result. Square it and add to the square of the number block formed by the rest of the result. The sum is always divisible by 137.[/quote]
very interesting - is there a known function for what value of y (in terms of x) will be produced by this algorithm for an input of 137*x giving an output of 137*y?

and now a simpler thinking one (hopefully)... 1634 (8208, and 9474 share the property I'm thinking of)

113=1+2(5)(7)+2(3)(7)
It can be broken down into the identity 1 and another 2 numbers composed of
the first 4 primes, added together. :)

113=11^2-2^3
The difference of 2 primes each to prime powers.


What do these numbers do? How good are they in terms of effieciency?
What's the shortest possible number that does this? For other cases?

9944339837363531292423228262021848288785808173472787707916964636268
6766159545352585756550510903020807060500114131218171615194044938974


1504201181097969594939291998878685848382808978776757473727170796656
4636261606905855453512594847454434149383634332313039272625240221100

<EDIT> There's also a relationship involving the English language and the number of letters.

Special whole numbers
 

[QUOTE=nibble4bits]

What do these numbers do? How good are they in terms of effieciency?
What's the shortest possible number that does this? For other cases?

9944339837363531292423228262021848288785808173472787707916964636268
6766159545352585756550510903020807060500114131218171615194044938974


1504201181097969594939291998878685848382808978776757473727170796656
4636261606905855453512594847454434149383634332313039272625240221100

<EDIT> There's also a relationship involving the English language and the number of letters.[/QUOTE]
:unsure: Its 10days since you posted the above and no solutions so far. So how about breaking the spell Nibble4bits ?
Mally :coffee:

Hint: look at sequential pairs of numbers.
The spoilers are in order of how little the give away. Avoid #2 and #3 until working on #1 and the hint.

[spoiler] The theoretically most efficient sequence would have to be greater then a
hundred digits long and it's trivial to convert it into another sequence of the same
length that ties. If you allow it to wrap around then you can reduce
the length by one.
[/spoiler]


This spoiler should be enough. Second spoiler is another sequence:
[spoiler]
In base 3 what does this do?
00221221021020021101
How would you propose to reduce it's length?
What's the smallest sequence that does it's job?
What if you allowed a limited precentage of 'gaps' in the solution?
[/spoiler]

You'll not want to read this last spoiler before the first two. Here's a related number:
[spoiler]
Spaces added to unobfusicate it.
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
[/spoiler]


PS: Yes I know that about the relationship between 311 and 131 and 113. :)

Here's an attachment with QBASIC source code.

Special whole numbers
 

[QUOTE=nibble4bits]Hint: look at sequential pairs of numbers.
The spoilers are in order of how little the give away. Avoid #2 and #3 until working on #1 and the hint.

[spoiler] The theoretically most efficient sequence would have to be greater then a
hundred digits long and it's trivial to convert it into another sequence of the same
length that ties. If you allow it to wrap around then you can reduce
the length by one.
[/spoiler]


This spoiler should be enough. Second spoiler is another sequence:
[spoiler]
In base 3 what does this do?
00221221021020021101
How would you propose to reduce it's length?
What's the smallest sequence that does it's job?
What if you allowed a limited precentage of 'gaps' in the solution?
[/spoiler]

You'll not want to read this last spoiler before the first two. Here's a related number:
[spoiler]
Spaces added to unobfusicate it.
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
[/spoiler]


PS: Yes I know that about the relationship between 311 and 131 and 113. :)[/QUOTE]
:smile: You seem to be dealing in sequences. As you know there are thousands of these given in Sloanes.
Since you have not given another number I will take the liberty of presenting an important open ended number worth noting
It is 1.2020569031 595942854...
Hint: it actually has some use in Number theory.
Mally :coffee:

Special whole numbers
 

[QUOTE=nibble4bits]Here's an attachment with QBASIC source code.[/QUOTE]
:whistle: On opening your attachment my browser's anti virus software alerted me of a virus in your files. :surprised
Mally :coffee:

Oooooooooh you mean because it's a ZIP? Or because it has BASIC code in it so it's in a blanket filter?

I'm so clever I got a virus into a plain text file that only notepad knows how to load. j/k Load it in Edit.com or Notepad.exe to see if those 3 text files are safe. I bet you'll just say "stupid AV strikes again!" >:)

But it's funny that people think their AV tools are always right. Not that I'm saying you couldn't figure that out.


<Edit> Oh yeah, that's used in physics as well. It's this: Sum of n^-3 for n=0 to infinite.