Re: FindInstance for sum of primes

• To: mathgroup at smc.vnet.net
• Subject: [mg115539] Re: FindInstance for sum of primes
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 14 Jan 2011 06:16:42 -0500 (EST)

```Here are the first 100 sums of 11 consecutive primes:

Clear[f];
f[1] = Total@Prime@Range@11
f[n_] := f[n] = f[n - 1] + Prime[n + 10] - Prime[n - 1]
Array[f, {100}] // Timing

160

{0.0009, {160, 195, 233, 271, 311, 353, 399, 443, 491, 539, 583, 631,
677, 725, 779, 833, 883, 931, 979, 1025, 1081, 1139, 1197, 1253,
1313, 1367, 1423, 1483, 1543, 1607, 1673, 1727, 1787, 1843, 1901,
1951, 2011, 2077, 2141, 2203, 2263, 2323, 2383, 2443, 2507, 2573,
2643, 2703, 2757, 2811, 2865, 2925, 2993, 3063, 3125, 3185, 3253,
3321, 3397, 3469, 3541, 3617, 3691, 3757, 3825, 3895, 3967, 4033,
4097, 4159, 4229, 4297, 4369, 4435, 4501, 4565, 4631, 4699, 4763,
4825, 4883, 4943, 5009, 5069, 5135, 5199, 5265, 5337, 5403, 5483,
5567, 5657, 5741, 5823, 5903, 5981, 6065, 6149, 6227, 6305}}

Bobby

On Thu, 13 Jan 2011 02:27:33 -0600, Peter Pein <petsie at dordos.net> wrote:

> 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
>

--
DrMajorBob at yahoo.com

```

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