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From: Ludovicus on 26 Jun 2010 07:55
As the primes are of the form 6n+1 and 6n+5 we can build 3 tables of
double entry
with the combinations: 6n-1, 6m-1 ; 6n-1, 6m+ 1 ; 6n - 1, 6m -
1 . n=1,2,3...m=1,2,3...
Example: (For the first table)

5 11 17 23 29 35 40 .....6n -1

5 10 16 22 28 34 40

11 16 22 28 34 40

17 22 28 34 40
..
..
6m-1

In this manner, the diagonals contains all the even numbers less than
46 formed by
the adding of pairs of primes less than 5*5.(That is, without
utilizing the composites
of the arithmetic progressions)

If we form the 36 tables from combinations of the 8 forms: 30n+ 1, 30n
+7, 30n+11
30n+13, 30n+17, 30n+19, 30n+23, 30n+29 then we will have the all the
even numbers
less than 94 formed by the adding of primes less than 7*7. (94 = 2*47)

And so on. The next are the 48 forms 210+1, 210+11.....,210+199.
Ludovicus
From: Ludovicus on 26 Jun 2010 08:29
On Jun 26, 7:55 am, Ludovicus <luir...(a)yahoo.com> wrote:
> As the primes are of the form 6n+1 and  6n+5 we can build 3 tables of
> double entry
> with the combinations: 6n-1, 6m-1 ;  6n-1, 6m+ 1  ;  6n - 1, 6m -
> 1  .  n=1,2,3...m=1,2,3...
> Example: (For the first table)
>
>         5    11    17    23    29    35    40 .....6n -1
>
>  5   10    16    22    28    34    40
>
> 11  16    22    28    34    40
>
> 17   22   28    34    40
> .
> .
> 6m-1
>
> In this manner, the diagonals contains all the even numbers less than
> 46 formed by
> the adding of pairs of primes less than 5*5.(That is, without
> utilizing the composites
> of the arithmetic progressions)
>
> If we form the 36 tables from  combinations of the 8 forms: 30n+ 1, 30n
> +7, 30n+11
> 30n+13, 30n+17, 30n+19, 30n+23, 30n+29 then we will have the all the
> even numbers
> less than 94 formed by the adding of primes less than 7*7. (94 = 2*47)
>
> And so on. The next are the 48 forms 210+1, 210+11.....,210+199.
> Ludovicus

CORRIGED VERSION:
As the primes are of the form 6n+1 and 6n+5 we can build 3 tables of
double entry> with the combinations: 6n-1, 6m-1 ; 6n-1, 6m+1 ; 6n-1,
6m-1
n=1,2,3...m=1,2,3...
Example: (For the first table)

5 11 17 23 29 35 40 .....6n-1

5 10 16 22 28 34 40

11 16 22 28 34 40

17 22 28 34 40
.
.
6m-1

In this manner, the diagonals contains all the even numbers less than
46 formed by the adding of pairs of primes less than 5*5.(That is,
without
utilizing the composites of the arithmetic progressions) (46 = 2*23)

If we form the 66 tables from combinations of the 8 forms: 30n+ , 30n
+7
30n+11 30n+13, 30n+17, 30n+19, 30n+23, 30n+29 plus the former 3
then we will have the all the even numbers less than 94 formed by the
adding of primes less than 7*7. (94 = 2*47)

And so on. The next are the 48 forms 210+1, 210+11.....,210+199 plus
the
former 11. That is 59*60/2 tables = 1770 producing the 226 even
number
resulting by the adding of the primes < 11*11. (226 = 2*113)
Ludovicus

From: curious george on 26 Jun 2010 14:27

"Ludovicus" <luiroto(a)yahoo.com> wrote in message
news:3e331e6c-8112-47f0-bde6-f88f17425e62(a)b35g2000yqi.googlegroups.com...
On Jun 26, 7:55 am, Ludovicus <luir...(a)yahoo.com> wrote:
> As the primes are of the form 6n+1 and 6n+5 we can build 3 tables of
> double entry
> with the combinations: 6n-1, 6m-1 ; 6n-1, 6m+ 1 ; 6n - 1, 6m -
> 1 . n=1,2,3...m=1,2,3...
> Example: (For the first table)
>
> 5 11 17 23 29 35 40 .....6n -1
>
> 5 10 16 22 28 34 40
>
> 11 16 22 28 34 40
>
> 17 22 28 34 40
> .
> .
> 6m-1
>
> In this manner, the diagonals contains all the even numbers less than
> 46 formed by
> the adding of pairs of primes less than 5*5.(That is, without
> utilizing the composites
> of the arithmetic progressions)
>
> If we form the 36 tables from combinations of the 8 forms: 30n+ 1, 30n
> +7, 30n+11
> 30n+13, 30n+17, 30n+19, 30n+23, 30n+29 then we will have the all the
> even numbers
> less than 94 formed by the adding of primes less than 7*7. (94 = 2*47)
>
> And so on. The next are the 48 forms 210+1, 210+11.....,210+199.
> Ludovicus

CORRIGED VERSION:
As the primes are of the form 6n+1 and 6n+5 we can build 3 tables of
double entry> with the combinations: 6n-1, 6m-1 ; 6n-1, 6m+1 ; 6n-1,
6m-1
n=1,2,3...m=1,2,3...
Example: (For the first table)

5 11 17 23 29 35 40 .....6n-1

5 10 16 22 28 34 40

11 16 22 28 34 40

17 22 28 34 40
.
.
> 6m-1

> In this manner, the diagonals contains all the even numbers less than
> 46 formed by the adding of pairs of primes less than 5*5.(That is,
> without
> utilizing the composites of the arithmetic progressions) (46 = 2*23)

> If we form the 66 tables from combinations of the 8 forms: 30n+ , 30n
> +7
> 30n+11 30n+13, 30n+17, 30n+19, 30n+23, 30n+29 plus the former 3
> then we will have the all the even numbers less than 94 formed by the
> adding of primes less than 7*7. (94 = 2*47)

> And so on. The next are the 48 forms 210+1, 210+11.....,210+199 plus
> the
> former 11. That is 59*60/2 tables = 1770 producing the 226 even
> number
> resulting by the adding of the primes < 11*11. (226 = 2*113)
> Ludovicus

not sure I understand why you needed to consider the 30n+....and the 210+
etc.
the 6n+/-1 has all the primes ( except 2 and 3) so you should be able to
show that using only these two progressions you can produce all the even
numbers.



From: Ludovicus on 27 Jun 2010 04:33
On Jun 26, 2:27 pm, "curious george" <bu...(a)bunch.net> wrote:

>
> not sure I understand why you needed to consider the 30n+....and the 210+
> etc.
> the 6n+/-1 has all the primes ( except 2 and 3) so you should be able to
> show that using only these two progressions you can produce all the even
> numbers.- Hide quoted text -
>
Remember that I must to evite the use of composites in the sums,
because
someone can argument that a diagonal was produced with composite only.
The composites in 6n+1 begin in 25 but in 30n+1 in 49

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