Primenumbers
Yahoo Groups

Two Sets of Consecutive Primes and their Sum of Digits Connection 

===============================================
w_sindelar@juno.com     Message 1 of 12  Aug 26, 2016
-----------------------------------------------
One, there exist 2 sets of n consecutive positive odd primes A=(P1<P2<P3...P(n)) and B=(Q1<Q2<Q3...Q(n)), such that the sum of the digits of A(n) equals B(n).

For example, consider these two sets of 5 consecutive odd primes: A=(402131, 402133, 402137, 402139, 402197) and B=(11, 13, 17, 19, 23). The sum of the digits of any member of set A, equals a corresponding member of set B.

Here is the next one; A=(1102313, 1102333, 1102337, 1102393, 1102397). B=(11, 13, 17, 19, 23).

A random one; A=(3678221, 3678223, 3678247, 3678287, 3678289). B=(29, 31, 37, 41, 43).

Here is the next one;  A=(4775231, 4775233, 4775293, 4775297, 4775299). B=(29, 31, 37, 41, 43).

I noticed that either one or the other, or both sets of all i tested, contained twin primes. Is this significant?

I found such sets with 2, 3, 4 primes, but so far none for n=6. Can anyone?

Thanks folks.

Bill Sindelar 

____________________________________________________________
hooch.net (Sponsored by Content.Ad)

===============================================
Jens Kruse Andersen     Message 2 of 12  Aug 26, 2016
-----------------------------------------------
Bill Sindelar wrote:
> One, there exist 2 sets of n consecutive positive odd primes
> A=(P1<P2<P3...P(n)) and B=(Q1<Q2<Q3...Q(n)), such that the
> sum of the digits of A(n) equals B(n).

> I found such sets with 2, 3, 4 primes, but so far none for n=6. Can anyone?

The first case for n=6:
6340271501, 6340271521, 6340271527, 6340271657, 6340271677, 6340271699
Digit sums: 29, 31, 37, 41, 43, 47

The first 10 cases all have those digit sums and start at:
6340271501, 7201850501, 16980112001, 20016507701, 22730027501,
24701360501, 100421445701, 103379004101, 108053035301, 121801644101

-- 
Jens Kruse Andersen
===============================================
Jens Kruse Andersen     Message 3 of 12  Aug 26, 2016
-----------------------------------------------
I wrote:
> The first case for n=6:
> 6340271501, 6340271521, 6340271527, 6340271657, 6340271677, 6340271699

This is a known problem.
https://oeis.org/A239790 shows 6340271501 was found by Giovanni Resta in 2014.
He says: "a(7), if exists, is larger than 2*10^14."

-- 
Jens Kruse Andersen
===============================================
Chroma     Message 4 of 12  Aug 27, 2016
-----------------------------------------------
'w_sindelar@...' w_sindelar@... [primenumbers] write:

> One, there exist 2 sets of n consecutive positive odd primes A=(P1<P2<P3...P(n)) and B=(Q1<Q2<Q3...Q(n)), such that the sum of the digits of A(n) equals B(n).
>
> For example, consider these two sets of 5 consecutive odd primes: A=(402131, 402133, 402137, 402139, 402197) and B=(11, 13, 17, 19, 23). The sum of the digits of any member of set A, equals a corresponding member of set B.
>

{{1811, 11}, {1831, 13}, {1871, 17}, {1873, 19}, {1877, 23}},
{{240041, 11}, {240043, 13}, {240047, 17}, {240049, 19}, {240089, 23}}

> Here is the next one; A=(1102313, 1102333, 1102337, 1102393, 1102397). B=(11, 13, 17, 19, 23).

The next is {{1022303, 11}, {1022341, 13}, {1022381, 17}, {1022383, 19}, {1022387, 23}}

> I found such sets with 2, 3, 4 primes, but so far none for n=6. Can anyone?

{{102401111, 11}, {102401113, 13}, {102401171, 17}, {102401263, 19}, {102401357, 23}, 
{102401399, 29}}

-- 
marian otremba
===============================================
Jens Kruse Andersen     Message 5 of 12  Aug 27, 2016
-----------------------------------------------
Chroma wrote:
> {{1811, 11}, {1831, 13}, {1871, 17}, {1873, 19}, {1877, 23}},

{1823, 14} and three other primes break it.

> {{240041, 11}, {240043, 13}, {240047, 17}, {240049, 19}, {240089, 23}}

{240059, 20} and one other prime breaks it.

> The next is {{1022303, 11}, {1022341, 13}, {1022381, 17}, {1022383, 19}, 
> {1022387, 23}}

{1022377, 22} breaks it.

> {{102401111, 11}, {102401113, 13}, {102401171, 17}, {102401263, 19},
> {102401357, 23}, {102401399, 29}}

{102401147, 20} and eight other primes break it.

None of your sets had consecutive primes.

-- 
Jens Kruse Andersen
===============================================
Chroma     Message 6 of 12  Aug 27, 2016
-----------------------------------------------
'Jens Kruse Andersen' jens.k.a@... [primenumbers] pisze:
>
> The first case for n=6:

   first ?

> 6340271501, 6340271521, 6340271527, 6340271657, 6340271677, 6340271699
> Digit sums: 29, 31, 37, 41, 43, 47

  {{{3775601, 29}, {3775621, 31}, {3775663, 37}, {3775757, 41}, {3775777, 43},{3775799, 47}},
{{7194701, 29}, {7194703, 31}, {7194727, 37}, {7194857, 41}, {7194877, 43}, {7194899,
47}},
{{15367601, 29}, {15367621, 31}, {15367717, 37}, {15367739, 41}, {15367777, 43}, 
{15367799, 47}},
{{17857001, 29}, {17857111, 31}, {17857153, 37}, {17857193, 41}, {17857249, 43}, 
{17857289, 47}},
{{20659601, 29}, {20659603, 31}, {20659627, 37}, {20659757, 41}, {20659759, 43}, 
{20659799, 47}}}  etc

-- 
marian otremba
===============================================
Jens Kruse Andersen     Message 7 of 12  Aug 27, 2016
-----------------------------------------------
Chroma wrote:
>  {{{3775601, 29}, {3775621, 31}, {3775663, 37}, {3775757, 41},
> {3775777, 43},{3775799, 47}},
>  {{7194701, 29}, {7194703, 31}, {7194727, 37}, {7194857, 41},
> {7194877, 43}, {7194899, 47}},
>  {{15367601, 29}, {15367621, 31}, {15367717, 37}, {15367739, 41},
> {15367777, 43}, {15367799, 47}},
>  {{17857001, 29}, {17857111, 31}, {17857153, 37}, {17857193, 41},
> {17857249, 43}, {17857289, 47}},
>  {{20659601, 29}, {20659603, 31}, {20659627, 37}, {20659757, 41},
> {20659759, 43}, {20659799, 47}}}  etc

None of these sets are close to being consecutive primes.
They are for example broken by these:
{3775619, 38}, {7194731, 32}, {15367607, 35}, {17857013, 32}, {20659619, 38}

Maybe your program ignores primes with a composite digit sum.
The primes have to be consecutive among all primes and
not just among primes with a prime digit sum.

-- 
Jens Kruse Andersen
===============================================
Chroma     Message 8 of 12  Aug 27, 2016
-----------------------------------------------
'Jens Kruse Andersen' jens.k.a@... [primenumbers] pisze:
> Chroma wrote:
>> {{1811, 11}, {1831, 13}, {1871, 17}, {1873, 19}, {1877, 23}},
>
> {1823, 14} and three other primes break it.
>
> {240059, 20} and one other prime breaks it.
>
> {1022377, 22} breaks it.
>
> {102401147, 20} and eight other primes break it.
>
> None of your sets had consecutive primes.

My mistake.
I calculated the sum of digits of prime numbers and then selected the ones that are the prime. Error !!!. They 
are not in order.

Sorry for the confusion

-- 
marian otremba
===============================================
Chroma     Message 9 of 12  Aug 27, 2016
-----------------------------------------------
Jens Kruse Andersen wrote:
>
> None of these sets are close to being consecutive primes.
> They are for example broken by these:
> {3775619, 38}, {7194731, 32}, {15367607, 35}, {17857013, 32}, {20659619, 38}
>
> Maybe your program ignores primes with a composite digit sum.
> The primes have to be consecutive among all primes and
> not just among primes with a prime digit sum.
>
You're right
I calculated the sum of digits of prime numbers and then selected the ones that are the prime. Error!. They 
are not in order.
Sorry for the confusion

-- 
marian otremba
===============================================
Jens Kruse Andersen     Message 10 of 12  Aug 27, 2016
-----------------------------------------------
Bill Sindelar wrote:
> One, there exist 2 sets of n consecutive positive odd primes
> A=(P1<P2<P3...P(n)) and B=(Q1<Q2<Q3...Q(n)), such that the
> sum of the digits of A(n) equals B(n).

The first case for n=7 looks hard due to the required number
of composites between the 7 primes in A. Here is a large case:

10^111+11*10^76+10^63+d, for d = 1, 111, 331, 351, 373, 429, 487
Digit sums: 5, 7, 11, 13, 17, 19, 23

-- 
Jens Kruse Andersen
===============================================
Jens Kruse Andersen     Message 11 of 12  Aug 27, 2016
-----------------------------------------------
Bill Sindelar wrote:
> One, there exist 2 sets of n consecutive positive odd primes
> A=(P1<P2<P3...P(n)) and B=(Q1<Q2<Q3...Q(n)), such that the
> sum of the digits of A(n) equals B(n).

I wrote:
> 10^111+11*10^76+10^63+d, for d = 1, 111, 331, 351, 373, 429, 487
> Digit sums: 5, 7, 11, 13, 17, 19, 23

That was the first n=7 case with digit sums 5 to 23.

No primes above 3 have digit sum 3.

The first n=7 with digit sums 7 to 29:
2000100000000000000100000000001010000000000000 + d,
for d = 1, 41, 223, 533, 733, 827, 869

The first n=7 with digit sums 11 to 31:
101100010001001200110000 + d,
for d = 1, 111, 223, 351, 373, 379, 399

All sets of 7 consecutive primes starting from 13 to 5623 have
a difference of at least 24 between the first and last prime.
That means the smallest possible difference between the first
and last prime in A for these sets is 798, the difference
between primes ending in the digits 001 and 799.
So solutions for these sets must have at least 792 composites
in total between the 7 primes.
That makes it look hard to find the smallest overall solution
when there are lots of small numbers with those digit sums.

-- 
Jens Kruse Andersen
===============================================
Chroma     Message 12 of 12  Aug 30, 2016
-----------------------------------------------
'Jens Kruse Andersen' jens.k.a@... [primenumbers] wrote:

> The first case for n=6:
> 6340271501, 6340271521, 6340271527, 6340271657, 6340271677, 6340271699
> Digit sums: 29, 31, 37, 41, 43, 47
>
> The first 10 cases all have those digit sums and start at:
> 6340271501, 7201850501, 16980112001, 20016507701, 22730027501,
> 24701360501, 100421445701, 103379004101, 108053035301, 121801644101

  The next 11 cases all have those digit sums and start at:
152032069001, 214255401401, 221026780001, 301305324701, 320605320701,
400786101101, 405560121401, 414321741101, 422236202303, 442262203301,
452402031701

1 case have (13, 17, 19, 23, 29, 31) digit sums and start at:
211001042101

-- 
marian otremba