Main Page | Report this Page
Science Forum Index  »  Mathematics Forum  »  sum n consecutive primes...
Page 2 of 2    Goto page Previous  1, 2

sum n consecutive primes...
Author Message
Jim Ferry...
Posted: Thu Oct 29, 2009 10:22 am
Guest
On Oct 28, 8:10 pm, "christian.bau" <christian.... at (no spam) cbau.wanadoo.co.uk>
wrote:
[quote]Related problem: For n >= 3, is it possible to find a number X (n)
that is both the sum and the product of n consecutive primes? For odd
n it should be not much harder to find (a bit harder because the
numbers involved are bigger). For even n, X (n) would have to be the
product of the _first_ n primes; so ind (n) = 1 in your solution. It
seems possible, but not very likely that a solution exists.

Playing a bit with heuristics:
Sum of 20 consecutive primes = product of 20 consecutive primes and
other primes -> easy to find.
Sum of 20 consecutive primes = product of 20 consecutive primes and
one more prime -> easy to find.
Sum of 20 consecutive primes = product of 20 first primes and one more
prime -> easy to find.
Sum of 20 consecutive primes = product of 20 or more consecutive
primes -> solutions are awfully hard to find.
[/quote]
Yes, this last one is an appealing problem, isn't it?
For example, it is nice that

11083 + 11087 + 11093 = 29*31*37,

or taking a case where a sum of 11 consecutive primes equals
a product of 11 consecutive primes,

216819892656221844131 + 216819892656221844133 +
216819892656221844139 + 216819892656221844169 +
216819892656221844307 + 216819892656221844331 +
216819892656221844347 + 216819892656221844373 +
216819892656221844397 + 216819892656221844401 +
216819892656221844421 67*71*73*79*83*89*97*101*103*107*109.

I wrote a program that takes increasingly
large candidate products and tests whether
there is a corresponding sum. Once it finds
five for a given n (the number of terms and
of factors), it stops using that n.

Here are the results of the program for
candidate products <= 8.55 * 10^103. It
took about 5 and a half hours. Results
were found for all odd n up through 25.
No results for even n were found (as
Christian points out, there is only one
candidate for each even n).

3 primes beginning with 11083
3 primes beginning with 2522057
3 primes beginning with 3560329
3 primes beginning with 9961421
3 primes beginning with 17293891
5 primes beginning with 555136752211
5 primes beginning with 2189102754909767
5 primes beginning with 6257081541269593
5 primes beginning with 8772225624149857
7 primes beginning with 7448535640735789
5 primes beginning with 20088362815834091
7 primes beginning with 141186655296130186657
11 primes beginning with 216819892656221844131
7 primes beginning with 60474807531776080924421
7 primes beginning with 107376274838007614601451
7 primes beginning with 8325615535695820235438171
9 primes beginning with 3484361319642920210271255507593
9 primes beginning with 78524120926727671771849976574513799
9 primes beginning with 3225708951152760224122644657197991241
9 primes beginning with 1197346152001756308791185325644857814199
9 primes beginning with 2748438901956435665082194599208405196751
11 primes beginning with 3112470056249097886304146985887140434683
11 primes beginning with 4604974935312951555626941275228083672957249
11 primes beginning with 12438909066251099275134302932635290752067413
11 primes beginning with
4831366125798491725766011414588665196476295583
13 primes beginning with
1808428663367515289053240288029870491511801302236142987
13 primes beginning with
589951208764060487330371783916854858390791939074455619239
13 primes beginning with
897924499092424257538219131877992876986550413524779985741
13 primes beginning with
2283106937031781509734806361833463157166699184607839118241
13 primes beginning with
28061533826186167017021127794471885758322546214655054770477
17 primes beginning with
9785006399109340879013256874616886354795520907037650463250430001893
15 primes beginning with
29078571394731776342443354126820734140155865170701085974267234571782657
15 primes beginning with
60072065163697224078306204701426643378780324165257271038398216308215353
15 primes beginning with
1321236339141292587740704764687378820214762385828852043312395417192634327
17 primes beginning with
2873216733612173842938072842675323149344746567084839209905876179862288199
15 primes beginning with
13496497478906418817186145250236666764058040718795841531181426429222469861
15 primes beginning with
118100177980841625516870911981522549576302388162737963551452979605616632059
17 primes beginning with
126484670661942269077177683188880151273371212438382900182347270545692665651453
21 primes beginning with
1125112337359143700021131708732871660802357010093542985845576342285073251235473
19 primes beginning with
2132759828550742498685066290824061285211489253263883046534515463344137569441423
17 primes beginning with
1198110630651125218157671663020047075451118091588201612761074887370468229098493133
17 primes beginning with
4180243301810820819646222439179052582963521139535037357110460003769072341286094149
19 primes beginning with
511893674858278714243549448706418619052641221985312207979314594713137784775382433193227
19 primes beginning with
190089539568552495146690644107640591582891762051521573856168142050912709302011708240957583
19 primes beginning with
74474568610606744006005724984987459049014253668590958465069031449152730286140247781268822267
19 primes beginning with
2871876266794999712288889548329344912596678105882896491524481372556996004797390909382839812730871
21 primes beginning with
5134878353748749608697474943557712564122877999370453170730042036379220140173664813048977905721453
21 primes beginning with
8364913822010080446224656837877585356969348354045469429188043539748343432025304371330705112392201
23 primes beginning with
2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490138629
21 primes beginning with
4429973166173553607958803008931422186300341493580464279101511020793700202888441501123980680475213813
25 primes beginning with
29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628028993

Here are just the first hits for each n, with
the n consecutive prime factors written out:

3 pbw 11083
{29, 31, 37}

5 pbw 555136752211
{293, 307, 311, 313, 317}

7 pbw 7448535640735789
{229, 233, 239, 241, 251, 257, 263}

9 pbw 3484361319642920210271255507593
{3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191}

11 pbw 216819892656221844131
{67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109}

13 pbw 1808428663367515289053240288029870491511801302236142987
{18121, 18127, 18131, 18133, 18143, 18149, 18169, 18181,
18191, 18199, 18211, 18217, 18223}

15 pbw
29078571394731776342443354126820734140155865170701085974267234571782657
{59629, 59651, 59659, 59663, 59669, 59671, 59693, 59699,
59707, 59723, 59729, 59743, 59747, 59753, 59771}

17 pbw
9785006399109340879013256874616886354795520907037650463250430001893
{10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301,
10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357,
10369}

19 pbw
2132759828550742498685066290824061285211489253263883046534515463344137569441423
{15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461,
15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551,
15559, 15569, 15581}

21 pbw
1125112337359143700021131708732871660802357010093542985845576342285073251235473
{5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029,
6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101,
6113}

23 pbw
2642367271219280209932806149780912734847221137377333153803807214361244928264753653400424464490138629
{24007, 24019, 24023, 24029, 24043, 24049, 24061, 24071,
24077, 24083, 24091, 24097, 24103, 24107, 24109, 24113,
24121, 24133, 24137, 24151, 24169, 24179, 24181}

25 pbw
29633546567618111878662501509234443150616537140595704746628589348284103223057398662776857001628028993
{11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831,
11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909,
11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969,
11971}

Finally, here is the Mathematica code:

init[] := (
ne = 4;
vale = 2*3*5*7;
nomin = 3;
nomax = nomin;
Clear[valo, ipmin, nTally];
valo[n_] := Product[Prime[j], {j, 2, n + 1}];
ipmin[_] := 2;
nTally[_] := 0;
talMax = 5;
nDone = {};
);
gete[] := {vale, ne};
geto[n_] := {valo[n], n};
itere[] := (ne += 2; vale *= Prime[ne - 1] Prime[ne]);
itero[n_] := {ipmin[n]++;
valo[n] = Product[Prime[j], {j, ipmin[n], ipmin[n] + n - 1}]};
nextCandidate[] := Module[{val, n, oddns},
oddns = Complement[Range[nomin, nomax, 2], nDone];
{val, n} = Sort[Prepend[geto / at (no spam) oddns, gete[]]][[1]];
If[EvenQ[n], itere[], itero[n]];
If[n == nomax, nomax += 2];
{val, n}
];
makeChain[v_, n_] : Rest[NestList[NextPrime, Floor[v/n - n Log[v/n]/2], n]];
iterChain[ch_, dir_] : If[dir > 0, Append[Rest[ch], NextPrime[Last[ch]]],
If[dir < 0, Prepend[Drop[ch, -1], -NextPrime[-First[ch]]], ch]];
validQ[v_, n_] := Module[{ch, dir, dirOld},
dir = Sign[v - Plus at (no spam) at (no spam) (ch = makeChain[v, n])];
While[dir = Sign[v - Plus at (no spam) at (no spam) (ch = iterChain[ch, dirOld = dir])];
dir*dirOld > 0];
{dir == 0, ch[[1]]}
];
find[itmax_] := Module[{v, n, vq},
init[];
Do[
{v, n} = nextCandidate[];
vq = validQ[v, n];
If[vq[[1]],
nTally[n]++;
If[nTally[n] >= talMax, AppendTo[nDone, n];
If[n == nomax, nomax += 2]];
Print[n, " primes beginning with ", vq[[2]]]],
{itmax}];
];
check[v_, n_] : FactorInteger[Plus at (no spam) at (no spam) NestList[NextPrime, v, n - 1]][[All, 1]];
Back to top  
Jim Ferry...
Posted: Thu Oct 29, 2009 10:35 am
Guest
On Oct 29, 3:09 pm, jm bergot <thekingfi... at (no spam) yahoo.ca> wrote:
[quote]What a stupendous amount of computational derring-do
by "sizzlin' laptop" Ferry!  All this should be perhaps
passed on to OEIS?

It is fortunate that I did not refer to two items at
primepuzzles.net which have ONLY two solutions listed.

It is equally fortunate that I did not suggest extending
A154598 at OEIS.

ThanXXX again to the vigorous electrons in Ferry's
sizlin' laptop.
[/quote]
Well, those electrons are resting now. Absolutely
motionless. But dang! Where are they?

2,29,293,229,3119,67,18121,59629,10247,15391,5903,
24007,11783,... makes kind of a nice sequence, I guess
(cf. what I posted a few minutes ago).
Back to top  
Gerry Myerson...
Posted: Thu Oct 29, 2009 6:12 pm
Guest
In article
<91a5b631-b007-4023-b9c3-12a89e3e9369 at (no spam) s21g2000prm.googlegroups.com>,
Jim Ferry <corklebath at (no spam) hotmail.com> wrote:

[quote]On Oct 29, 3:09 pm, jm bergot <thekingfi... at (no spam) yahoo.ca> wrote:
What a stupendous amount of computational derring-do
by "sizzlin' laptop" Ferry!  All this should be perhaps
passed on to OEIS?

It is fortunate that I did not refer to two items at
primepuzzles.net which have ONLY two solutions listed.

It is equally fortunate that I did not suggest extending
A154598 at OEIS.

ThanXXX again to the vigorous electrons in Ferry's
sizlin' laptop.

Well, those electrons are resting now. Absolutely
motionless. But dang! Where are they?
[/quote]
That reminds me - I've defined a quantum group to be one in which
you can know the order of any subgroup, or its index, but not both.

--
Gerry Myerson (gerry at (no spam) maths.mq.edi.ai) (i -> u for email)
Back to top  
Bill Taylor...
Posted: Thu Oct 29, 2009 7:39 pm
Guest
Gerry Myerson <ge... at (no spam) maths.mq.edi.ai.i2u4email> wrote:

[quote]That reminds me - I've defined a quantum group to be one in which
you can know the order of any subgroup, or its index, but not both.
[/quote]
COOL !!

Here is an even more quantummy thing, in plain old set theory.

One is given a set, and told that it is (U x V) union (V X U),
for some sets U and V.

Then it is easily seen that, from this, one can reconstruct
both U and V, but CANNOT TELL WHICH WAS WHICH!

This example was told to me by Thomas Forster, who says
it is a standard type of example for constructing models
of set theories that have various amusing properties.

But it has a distinctly quantummy feel to me!

-- Bi-natured Bill

** They travel as waves but arrive as particles.
Back to top  
}niaM olleH dlroW ediW beW...
Posted: Thu Oct 29, 2009 8:05 pm
Guest
On Oct 30, 1:39 am, Bill Taylor <w.tay... at (no spam) math.canterbury.ac.nz>
wrote:
[quote]Gerry Myerson <ge... at (no spam) maths.mq.edi.ai.i2u4email> wrote:
That reminds me - I've defined a quantum group to be one in which
you can know the order of any subgroup, or its index, but not both.

COOL !!

Here is an even more quantummy thing, in plain old set theory.

One is given a set, and told that it is (U x V) union (V X U),
for some sets U and V.

Then it is easily seen that, from this, one can reconstruct
both U and V, but CANNOT TELL WHICH WAS WHICH!

This example was told to me by Thomas Forster, who says
it is a standard type of example for constructing models
of set theories that have various amusing properties.

But it has a distinctly quantummy feel to me!

-- Bi-natured Bill

** They travel as waves but arrive as particles.
[/quote]
"can" (and any subsequent words) was ignored
Results 1 - 3 of about 4 for On Oct 30, 1:39 am, Bill Taylor
<w.tay... at (no spam) math.canterbury.ac.nz> wrote:> Gerry Myerson
<ge... at (no spam) maths.mq.edi.ai.i2u4email> wrote:> > That reminds me - I've
defined a quantum group to be one in which> > you can know the order
of any subgroup, or its index, but not both.>> COOL !!>> Here is an
even more quantummy thing, in plain old set theory.>> One is given a
set, and told that it is (U x V) union (V X U),> for some sets U and
V.>> Then it is easily seen that, from this, one can reconstruct> both
U and V, but CANNOT TELL WHICH WAS WHICH!>> This example was told to
me by Thomas Forster, who says> it is a standard type of example for
constructing models> of set theories that have various amusing
properties.>> But it has a distinctly quantummy feel to me!>> -- Bi-
natured Bill>> ** They travel as waves but arrive as particles.. (0.87
seconds)
Search Results

Results include your SearchWiki notes for On Oct 30, 1:39 am,
Bill Taylor <w.tay... at (no spam) math.canterbury.ac.nz> wrote:> Gerry Myerson
<ge... at (no spam) maths.mq.edi.ai.i2u4email> wrote:> > That reminds me - I've
defined a quantum group to be one in which> > you can know the order
of any subgroup, or its index, but not both.>> COOL !!>> Here is an
even more quantummy thing, in plain old set theory.>> One is given a
set, and told that it is (U x V) union (V X U),> for some sets U and
V.>> Then it is easily seen that, from this, one can reconstruct> both
U and V, but CANNOT TELL WHICH WAS WHICH!>> This example was told to
me by Thomas Forster, who says> it is a standard type of example for
constructing models> of set theories that have various amusing
properties.>> But it has a distinctly quantummy feel to me!>> -- Bi-
natured Bill>> ** They travel as waves but arrive as particles.. Share
these notes
Copy and paste this link into an email or IM:

See a preview of the shared page
1.
mm-4584 - glk's files on science AND religion
[quote]Han de Bruijn One can certainly define such a function from N
to {N} in ZFC, ...... Jesse F. Hughes It is a brilliant proof you, you[/quote]
math haters! ...... Bill Taylor Exactly. ---David Hobby === Subject:
Re: Set existence Originator: ...... Gerry Myerson
(gerry at (no spam) maths.mq.edi.ai) (i -> u for email) === Subject: Re: ...
grahamkendall.net/Math/Math%20Newsgroups/mm-4584.txt - Cached -
Similar -
2.
Dehn invariant - NIU Math Department
If you search math reviews, looking for `Dehn invariant', you'll
get some of the key ... Gerry Myerson <gerry at (no spam) mpce.mq.edu.au> wrote: >
[quote]In article ... Bill Taylor <mathwft at (no spam) math.canterbury.ac.nz> wrote:
=>> So is it possible to ..... Group homomorphisms on $\scr P$ are
defined from their values on polyhedra, ...[/quote]
www.math.niu.edu/~rusin/known-math/98/sydler - Cached - Similar
-
3.
Math Forum Discussions
4), Re: Welcome to <sci.math>. These suggestions may help you.
[39] ... news.motzarella.org>, > Gerry Myerson
<ge... at (no spam) maths.mq.edi.ai.i2u4email> . ... On May 29, 3:58 am, Ken
Pledger <ken.pled... at (no spam) mcs.vuw.ac.nz> wrote: > > It can be shown
that ... Posted: Dec 14, 2008 9:32 PM, by: Bill Taylor Ken Pledger
<ken.pled. ...
mathforum.org/kb/search!execute.jspa?q=vuw - Cached - Similar -

You have removed results from this search. Hide them
Loading...
Back to top  
jm bergot...
Posted: Fri Oct 30, 2009 7:19 am
Guest
The latst explosion of yottaflops from "sizzlin' laptop"
Ferry's machine should also be zinged to OEIS.

How this led to quantum chitchat makes me wonder if the
proper adjective for 'quantum' should be either
'quantumoid' or 'quantumish' rather than 'quantummy',
which sounds like something you take for an upset
stomach.
Back to top  
Gerry Myerson...
Posted: Sun Nov 01, 2009 9:34 pm
Guest
In article
<0de89d78-6405-47c6-ba05-ce973916bc94 at (no spam) i12g2000prg.googlegroups.com>,
Bill Taylor <w.taylor at (no spam) math.canterbury.ac.nz> wrote:

[quote]Gerry Myerson <ge... at (no spam) maths.mq.edi.ai.i2u4email> wrote:

That reminds me - I've defined a quantum group to be one in which
you can know the order of any subgroup, or its index, but not both.

COOL !!

Here is an even more quantummy thing, in plain old set theory.

One is given a set, and told that it is (U x V) union (V X U),
for some sets U and V.

Then it is easily seen that, from this, one can reconstruct
both U and V, but CANNOT TELL WHICH WAS WHICH!
[/quote]
Is this really different from saying that given a binomial,
and being told it's U + V for some monomials U and V,
one can reconstruct both U and V, but cannot tell which
was which?

--
Gerry Myerson (gerry at (no spam) maths.mq.edi.ai) (i -> u for email)
Back to top  
master1729...
Posted: Mon Nov 02, 2009 11:15 am
Guest
[quote]In article
91a5b631-b007-4023-b9c3-12a89e3e9369 at (no spam) s21g2000prm.goog
legroups.com>,
Jim Ferry <corklebath at (no spam) hotmail.com> wrote:

On Oct 29, 3:09 pm, jm bergot
thekingfi... at (no spam) yahoo.ca> wrote:
What a stupendous amount of computational
derring-do
by "sizzlin' laptop" Ferry!  All this should be
perhaps
passed on to OEIS?

It is fortunate that I did not refer to two items
at
primepuzzles.net which have ONLY two solutions
listed.

It is equally fortunate that I did not suggest
extending
A154598 at OEIS.

ThanXXX again to the vigorous electrons in
Ferry's
sizlin' laptop.

Well, those electrons are resting now. Absolutely
motionless. But dang! Where are they?

That reminds me - I've defined a quantum group to be
one in which
you can know the order of any subgroup, or its index,
but not both.

--
Gerry Myerson (gerry at (no spam) maths.mq.edi.ai) (i -> u for
email)
[/quote]
surely your joking mr feynman.
Back to top  
 
    Display posts from previous:  
Page 2 of 2    Goto page Previous  1, 2
All times are GMT - 5 Hours
The time now is Sat Dec 12, 2009 4:25 am
Science Forum Index
 

Powered by phpBB & ML | Privacy Policy / Disclaimer



Contact Us