Re: Re: how to solve the integer equation Abs[3^x-2^y]=1
- To: mathgroup at smc.vnet.net
- Subject: [mg103157] Re: [mg103076] Re: how to solve the integer equation Abs[3^x-2^y]=1
- From: a boy <a.dozy.boy at gmail.com>
- Date: Wed, 9 Sep 2009 04:45:48 -0400 (EDT)
- References: <200909031110.HAA24198@smc.vnet.net> <h7tbeb$fs6$1@smc.vnet.net>
On Wed, Sep 9, 2009 at 10:03 AM, a boy <a.dozy.boy at gmail.com> wrote:
>
>
> On Wed, Sep 9, 2009 at 12:18 AM, Daniel Lichtblau <danl at wolfram.com>wrote:
>
>>
>> [...]
>>> For any integer k and 3^k, suppose 2^j is the closest to 3^k, Gap[k]=|
>>> 3^k-2^j| is the subtraction .
>>>
>>> Gap = Function[k, x = k*Log[2, 3]; Min[3^k - 2^Floor[x], 2^Ceiling
>>> [x] - 3^k]];
>>> Table[{i, Gap[i]}, {i, 1, 100}]
>>>
>>> Out[24]:={{1, 1}, {2, 1}, {3, 5}, {4, 17}, {5, 13}, {6, 217}, {7,
>>> 139}, {8,
>>> 1631}, {9, 3299}, {10, 6487}, {11, 46075}, {12, 7153},.....
>>> I find {Gap[i]} is not a increasing sequence. Suppose D is a strict
>>> decreasing sub sequence of {Gap[i]} .
>>> Q1: is the length of D always less than 3?
>>>
>>
>> I suspect it is straightforward to show that you cannot have three
>> consecutive decreases.
>>
>> As for getting any such subsequence, let's first define, for given
>> nonnegative integers mj and nj, the real value tj by
>>
>> | mj*log(2) / (nj*log(3)) | = 1 + tj
>>
>> The idea being, we want to find pairs {mj,nj} with corresponding tj very
>> small. In this setting, we have
>>
>> 2^mj - 3^nj = 3^nj * (3^(tj*nj)-1)
>>
>> So what we require is an increasing set m1, m2, m3 and corresponding n1,
>> n2, n3 such that the sequence 3^nj * (3^(tj*nj)-1) decreases. To first order
>> approximation, this value is tj*nj*3^nj.
>>
>> Can we have such trios? Perhaps naively, I think this would depend on
>> having "large" convergents somewhere in the continued fraction
>> representation of log(2)/log(3). But regardless, the answer is yes, we do
>> have such trios. Here is one such.
>>
>> In[48]:= {Gap[666], Gap[661], Gap[660]} // N
>>
>> 317 314
>> Out[48]= {1.930005508972960 10 , 6.328896257794369 10 ,
>>
>> 313
>> > 4.037250828437273 10 }
>>
> ~~~~~~~~~~~~~~~~~~~~
> I'm soooorry!
>
orderedlogs = Ordering[Table[Log[N[Gap[k]]], {k, 1, 5000}]] shows a trio.
It is 659,660,665! not 660,661,666.
Gap[665]< Gap[660]< Gap[659]
>
>> I found this using the code below.
>>
>> Gap[k_] := With[{x=k*Log[2, 3]}, Min[3^k-2^Floor[x], 2^Ceiling[x]-3^k]]
>> orderedlogs = Ordering[Table[Log[N[Gap[k]]], {k, 1, 5000}]]
>> orderdiffs = ListConvolve[{1,-1}, orderedlogs]
>>
>> Now just look for two consecutive negative signs:
>>
>> In[61]:= conseqs = Position[Partition[orderdiffs,2,1],
>> {a_,b_} /; a<0&&b<0]
>> Out[61]= {{659}, {1324}, {1989}, {2654}}
>>
>> It is reasonable to conjecture that there is an upper bound on these
>> decreasing subsequence lengths. If I up the size from 5000 to 10000
>> elements, I do not get further trios, which indicates it might be reasonable
>> to conjecture that the maximum length of decreasing gap subsequences is in
>> fact 3. But when I increas again to 20000, I get a sizeably larger set:
>>
>> Out[67]= {{659}, {1324}, {1989}, {2654}, {12935}, {13600}, {14265},
>> {14930}, {16925}, {17590}, {18255}, {18920}}
>>
>> Does this mean we might expect decreasing subsequences of length 4 or
>> larger? I do not know. One sign that would make me suspect a negative answer
>> is that the pattern near such trios is always the same.
>>
>> In[78]:= Map[orderdiffs[[#[[1]]-2;;#[[1]]+3]]&, conseqs]
>>
>> Out[78]= {
>> {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1},
>> {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1},
>> {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1},
>> {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1}, {1, 7, -5, -1, 2, 1}}
>>
>> So we have recurring gap undulations, in a manner of speaking.
>>
>>
>> -------------
>>> I have another question.
>>>
>>> Table[Abs[s2 * 2^m + s3 *3^n], {s2, {-1, 1}}, {s3, {-1, 1}}, {m, 0,
>>> 100}, {n, 0, 100}];
>>> Tally[Sort[Flatten[%]]]
>>>
>>> The result shows that 21 != 2^i+3^j or |2^i-3^j| and 53 can not be
>>> expressed as these form also.
>>> But 53= 2 * 3^3 - 1
>>> My another question is:
>>> Q2: Is any odd prime number p can be expressed as one of these forms:
>>> 1. 2^i + 3^j
>>> 2. 2^i - 3^j or 3^i - 2^j
>>> 3. 2^i * 3^j +1
>>> 4. 2^i * 3^j -1
>>>
>>> The answer to Q2 is true of false? How to prove or disprove it?
>>>
>>
>> Again I do not know the answer but my guess is it is false. Call the
>> values you cannot attain in your table (extended to infinity...) non-gaps. I
>> would expect the density of such nongaps to be far too large to recover them
>> all as numbers in the form 3 or 4 above.
>>
>> Again, this might be tied to behavior of the continued fraction of
>> log(2)/log(3).
>>
>> Daniel Lichtblau
>> Wolfram Research
>>
>
>
- References:
- how to solve the integer equation Abs[3^x-2^y]=1
- From: a boy <a.dozy.boy@gmail.com>
- how to solve the integer equation Abs[3^x-2^y]=1