- I was going to say your t numbers are denser than the triangular numbers, so

the case for 4 has no solutions, but this doesn't seem to be true.

I would have compared the asymtopic formulae, unfortunately EIS doesn't seem

to have one for the triangular numbers.

Jon Perry

perry@...

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-----Original Message-----

From: Mark Underwood [mailto:mark.underwood@...]

Sent: 29 August 2003 16:21

To: primenumbers@yahoogroups.com

Subject: [PrimeNumbers] The "twothree" numbers

Hi all

I would like to introduce the 'twothree' numbers, abbreviated as

the 't' numbers. A 't' number is any number which has not factors

other than 1 ,2 and 3. A 't' number is of the form (2^a)*(3^b) ,

where a and b are non negatve integers. The 't' numbers are:

1,2,3,4,6,8,9,12,16,18,24,27,32,36 and so on. The first number which

doesn't qualify as a 't' number is a prime, 5.

Now take any two 't' numbers and add them together. ie,

1+1 = 2 ; 1+2 = 3 ; 1+3 = 4 ; 1+4 = 5 ; 2+4 = 6. ; 3+4 = 7, etc.

It turns out that the first number greater than 1 which can't be

expressed as the sum of two 't' numbers at least once is the number

23. Upon some reflection we see that this number would have to be

prime.

Now take any three 't' numbers and add them together. ie,

1+1+1 = 3 ; 1+1+2 = 4 ; 1+1+3 = 5 ; 1+1+4 = 6 ; 1+2+4 = 7, etc.

The first number greater than 2 which can't be expressed as the sum

of three 't' numbers at least once is the number 431. As we would

expect, it is again a prime.

Now take any four 't' numbers and add them together. ie,

1+1+1+1 = 4; 1+1+1+2 = 5 ; 1+1+1+3 = 6, etc.

I have not been able to find the first number (which would be a

prime) which cannot be expressed as the sum of 4 't' numbers. I

suspect it is huge but I'm not sure of what order. And it would

certainly be a prime number.

Here are some figures that give an idea of the number of solutions.

It suffices to consider only prime numbers.

5 = 1+1+1+2.

7 = 1+1+1+4 = 1+1+2+3 = 1+2+2+2.

11= 1+1+1+8 = 1+1+3+6 = 1+2+2+6 = 1+2+4+4 = 1+3+3+4 = 2+2+3+4 =

2+3+3+3.

In summary, 5 has one solution, 7 has 3 solutions and 11 has 7

solutions, all expressed as (5,1) (7,3) (11,7). Here is a more

comprehensive list:

(5,1) (7,3) (11,7) (13,9) (17,13) (19,15) (23,19) (29,23) (31,24)

(37,30) (41,32) (43,34) (47,34) (53,36) (59,37) (61,40) (67,41)

(71,40) (73,42) (79,43) (83,45) (89,47) (97,48) (101,49) (103,50)

(107,50) (109,52) (113,51) (127,51) (131,49) (137,54) (139,56)

(149,53) (151,52) (157,56) (163,58) (167,53) (173,56) (179,56)

(181,59) (191,48) (193,56) (197,52) (199,55) (211,55) (223,48)

(227,58) (229,57) (233,58) (239,45) (241,56) (251,54) (257,59)

(263,55) (269,57) (271,57) (277,62) (281,65) (283,63) (293,57)

(307,71) (311,53) (313,67) (317,59) (331,67) (337,70) (347,62)

(349,59) (353,67) (359,53) (367,54) (373,57) (379,64) (383,45)

(389,54) (397,60) (401,58) (409,68) (419,60) (421,57) (431,36)

(433,61) (439,51) (443,59) (449,54) (457,61) (461,53) (463,52)

(467,57) (479,42) (487,51) (491,56) (499,64) (503,42) ...

The computation time gets way out of hand as the numbers get larger.

I tried computing the solutions for the single prime 100003 and after

eight hours running it has given 10 solutions so far, the most recent

being 243 + 432 + 16384 + 82944.

Interesting and somewhat satisfied my curiousity but I can't see that

this could lead anywhere useful.

Mark

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The Prime Pages : http://www.primepages.org/

Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ - In a message dated 30/08/03 08:04:45 GMT Daylight Time, jack@...

writes:

> The smallest number not expressible as a sum of 5 't' numbers

I have recoded Jack's elegant algorithm in Pascal, at the same time changing

> is the number 3448733 (37 * 83 * 1123).

>

> Here is the C program I used. If you have enough RAM, you can extend

> the search beyond the 20000000 that I searched to. This program should

> work unmodified for SLIM as high as 1400000000 (if you've got

> about 1.5 gigabytes of RAM). Go much beyond that and you'll have to

> solve some issues with arithmetic overflow.

>

his char array to a bit-array, and run it for SLIM = 2*10^9, requiring a mere

250Mb of RAM.

An answer has just popped out after 6 hrs on an Athlon XP2800+ (2.08GHz):-

1441896119

This is prime, as it happens.

Mike

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