Re: FindInstance for sum of primes
- To: mathgroup at smc.vnet.net
- Subject: [mg115517] Re: FindInstance for sum of primes
- From: Peter Pein <petsie at dordos.net>
- Date: Thu, 13 Jan 2011 03:27:33 -0500 (EST)
- References: <igisd3$pr9$1@smc.vnet.net>
On 12.01.2011 01:25, leigh pascoe wrote:
> Dear Mathgroup,
>
> 2011 is the sum of 11 consecutive primes. I want to check if any other
> years have a similar property.
>
> Now define
>
> f[n_, m_] := Sum[Prime[i], {i, n, n + m}];
> eq = Mod[f[n, m] - year, 1000] == 0
>
> and we see that
>
> In[64]:= Mod[f[37, 10] - 2011, 1000]
>
> Out[64]= 0
>
> but
>
> In[65]:= FindInstance[eq, {n, m, year}, Integers]
>
> During evaluation of In[65]:= FindInstance::exvar: The system contains a
> nonconstant expression i independent of variables {n,m,year}.>>
>
> Out[65]= FindInstance[Mod[-year + \!\(\*UnderoverscriptBox[\(\[Sum]\),
> \(i = n\), \(m + n\)]\(Prime[i]\)\), 1000] == 0, {n, m, year}, Integers]
>
> Apparently FindInstance doesn't like the dummy variable "i". How can we
> perform this search in Mathematica??
>
> LP
>
Hello Leigh,
I can't imagine how to solve your problem using Reduce/FindInstance, but
you can construct a table containing sums of k conecutive primes:
In[1]:= pSumTable =
Mod[
With[{pr = Prime[Range[1000]]},
Table[Total /@ Partition[pr, k, 1], {k, 100}]
],
10000];
say, you want to find 2011. Then use
In[2]:= LengthAndStart = Position[pSumTable, 2011, 2]
Out[2]= {{1, 305}, {3, 121}, {3, 551}, {9, 494}, {11, 37}, {11,
737}, {29, 170}, {35, 675}, {49, 342}, {49, 395}, {59, 16}, {61,
614}, {83, 241}, {91, 6}, {95, 606}}
{11, 37} is the case you mentioned in your post.
to verify the results (without the help of pSumTable):
In[3]:= Total[Prime[Range[#2, #2 + #1 - 1]]] & @@@ LengthAndStart
Out[3]= {2011, 2011, 12011, 32011, 2011, 62011, 32011, 182011, \
122011, 142011, 12011, 292011, 152011, 22011, 462011}
Cheers,
Peter