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Re: find two numbers a,b, such that a+b=5432 & LCM[a,b]=223020

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122165] Re: find two numbers a,b, such that a+b=5432 & LCM[a,b]=223020
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Tue, 18 Oct 2011 07:40:07 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j6e693$kef$1@smc.vnet.net> <201110050801.EAA07025@smc.vnet.net> <360851F3-7690-47A6-BC84-E9521100E6F4@mimuw.edu.pl> <A0A54467-5287-4372-9FA6-F0FD865B77D2@mimuw.edu.pl> <84B4CB1D-C8A7-4A5F-8303-9FEC113F7D16@mimuw.edu.pl> <1318773921.12784.YahooMailNeo@web43139.mail.sp1.yahoo.com> <201110162045.QAA19701@smc.vnet.net> <6371475F7C873A4CA2F9BBABC115837E0A5685CC@nycexch.local.atlanticphilanthropies.org> <201110171210.IAA26854@smc.vnet.net>

Actually, I have (sort of) changed my mind about it and I am glad that 
Harvey posted the "brute-force" approach. There are some interesting 
things that can be regarding about the performance of some of the 
proposed methods, including "brute-force". I am going to compare my 
solution, the solution sent by J=E1nos L=F6bb and Harvey's "brute-force" 
solution. The results seem instructive.

First let's try the original problem posted by Dimitris, then we will 
enlarge it, by multiplying all the integers by 10.

For the original problem we have:

my solution:

In[1]:= ls = Divisors[5432]; Timing[(Reduce[{a + b == 5432, a*b 
== 223020*#1}, {a, b},
      Integers] & ) /@ ls]
Out[1]= {0.0512979999999999, {False, False, False, False, False, 
False,
   (a == 1652 && b == 3780) || (a == 3780 && b == 1652), 
False,
   False, False, False, False, False, False, False, False}}


J=E1nos solution:

In[2]:= Timing[Reduce[{223020*(1/n + 1/m) == 5432}, {n, m}, 
Integers]]
Out[2]= {0.316161, (n == 59 && m == 135) || (n == 135 && m 
== 59)}

Harvey's solution

In[3]:= Timing[Select[Range[5432], LCM[#1, 5432 - #1] == 223020 & 
]]
Out[3]= {0.016298000000000035, {1652, 3780}}

Now, with larger numbers:

my solution

In[3]:= ls = Divisors[54320]; Timing[(Reduce[{a + b == 54320, 
a*b == 2230200*#1}, {a, b},
      Integers] & ) /@ ls]
Out[3]= {0.06448999999999971, {False, False, False, False, False, 
False,
   False, False, False, False, False, False, False, False, False,
   False, False, False, False, False,
   (a == 16520 && b == 37800) || (a == 37800 && b == 
16520),
   False, False, False, False, False, False, False, False, False,
   False, False, False, False, False, False, False, False, False,
   False}}

Janos's solution

In[5]:= Timing[Reduce[{2230200*(1/n + 1/m) == 54320}, {n, m}, 
Integers]]

Out[5]= {0.29640900000000014, (n == 59 && m == 135) ||
   (n == 135 && m == 59)}

Harvey's brute-force solution

In[6]:= Timing[Select[Range[54320], LCM[#1, 54320 - #1] == 2230200 
& ]]
Out[6]= {0.12470700000000012, {16520, 37800}}


Several things to observe. In the first round (the original problem) the 
brute-force solution wins easily, being about three times faster than my 
"mathematical" solution. Janos's solution is much slower. In the second 
round, the brute force solution is almost an order of magnitude slower - 
as one would expect, but the time taken by the two mathematical 
solutions is almost unchanged. My solution is now twice as fast as the 
brute-force one, Janos's is still slower but is catching up.

The reason why the brute-force method does so well in the original case 
but not so when the numbers become larger is obvious - although the 
mathematical problem obtained by multiplying everything by 10 is of 
about the same order of "mathematical difficulty" it is almost ten times 
harder for a brute-force approach.
It may be however less clear why my method is so much faster than 
Janos's. The simple reason is the Janos's method requires Reduce to 
solve a much harder mathematical problem.

Note that my solution does not really require any knowledge of 
Diophantine equations. First of all the divisors of the sum of a+b are 
computed - this can be done very fast  for numbers of this size (even by 
hand). Then the solution amounts to solving 16 (in the first case) 
ordinary quadratic equations and selecting the one that has integer 
solutions. This is something that could even be done "by hand" in a 
reasonable time.

Janos's solution, on the other hand, requires solving a quadratic 
Diophantine equation with two unknowns. This is a much harder problem - 
in fact I am not sure how Mathematica does it (there may be even some 
brute-force involved).

In any case, the relation between "mathematical complexity" of an 
approach and performance is not a straight forward one. The brute-force 
approach has essentially zero mathematical complexity - but that becomes 
a handicap for large problems of this kind. Janos's approach has a 
higher "mathematical complexity" than mine and this involves itself a 
significant penalty. It is hard for me to judge how the two approaches 
will compare when the numbers grow large because I do not know how 
Reduce solves this sort of Diophantine equations. My approach will 
certainly eventually suffer, as the number of divisors grows large, both 
form the difficulty of computing them and the growing number of 
quadratic equations that Reduce has to solve.

Andrzej Kozlowski



On 17 Oct 2011, at 14:10, Andrzej Kozlowski wrote:

> Indeed, but there did not seem (to me) to be much point in sending a

> "brute-force" solution to someone who specifically asks for something

> other than a brute-force solution.
>
> Andrzej
>
> On 16 Oct 2011, at 23:47, Harvey P. Dale wrote:
>
>> 	Here's a simple brute-force method:
>>
>> Select[Range[5431],LCM[#,5432-#]==223020&]
>>
>> 	Best,
>>
>> 	Harvey
>>
>> -----Original Message-----
>> From: Andrzej Kozlowski [mailto:akoz at mimuw.edu.pl]
>> Sent: Sunday, October 16, 2011 4:46 PM
>> To: mathgroup at smc.vnet.net
>> Subject: Re: find two numbers a,b, such that a+b=5432 &
>> LCM[a,b]=223020
>>
>> Well, I don't think Reduce can solve it "directly" but here is an 
easy
>> method that makes use of  Reduce and a tiny bit of elementary number
>> theory to solve it.
>>
>> Let g denote the GCD of a and b. Then we know that a b = 223020 g
>> (from the well known relationship between GCD and LCM). Hence we need 
to
>>
>> solve the equation Reduce[{a + b == 5432, a b == 223020 g}, 
{a,
>> b}, Integers] for all possible GCD candidates of a and b. Since the 
GCD
>> of a and b is always a divisor of their sum a+b == 5432 we can
>> simply test all possible divisors (there are only 16).
>>
>> ls = Divisors[5432]
>>
>> {1,2,4,7,8,14,28,56,97,194,388,679,776,1358,2716,5432}
>>
>> (Reduce[{a + b == 5432, a*b == 223020*#1}, {a, b}, Integers] 
& )
>> /@ ls
>>
>> {False, False, False, False, False, False,
>>  (a == 1652 && b == 3780) || (a == 3780 && b == 
1652),
>> False,
>>  False, False, False, False, False, False, False, False}
>>
>> Thus we see that the only possibility is that one of the numbers is 
1562
>>
>> and the other 3780.
>>
>> Andrzej Kozlowski
>>
>> On 16 Oct 2011, at 16:05, dimitris anagnostou wrote:
>>
>>> Hello Mr Kozlowski .
>>>
>>> This is from a post of mine to MathGroup that it hasn't appeared 
yet.
>>>
>>> This is taken from the recent book (2010): " Mathematica: A
>> Problem-Centered Approach" by R. Hazrat.
>>>
>>> "The sum of two positive integers is 5432 and their least common
>> multiple is
>>> 223020. Find the numbers."
>>>
>>> A solution:
>>>
>>> Do[If[LCM[i, 5432 - i] == 223020, Print[i, "  ", 5432 - i]], {i,
>> 1,  2718}]
>>> 1652  3780
>>>
>>> I wonder if we can solve the system of equations:
>>>
>>> a+b==5432&&LCM[a,b]==223020
>>>
>>> using codes that contain built in functions like Reduce.
>>>
>>> I guess this is not a trivial one because the so much powerful 
Reduce=
>
>> itself fails
>>>
>>> In[1]:= Reduce[{a + b == 5432, LCM[a, b] == 223020}, {a, ==

> b},
>> Integers]
>>>
>>> During evaluation of In[1]:= Reduce::nsmet:This system cannot be
>> solved with the methods available to Reduce. >>
>>>
>>> Out[1]= Reduce[{a + b == 5432, LCM[a, b] == 223020}, {a, ==

> b},
>> Integers]
>>>
>>> I would really appreciate your ideas.
>>>
>>> Best Regards
>>>
>>> Dimitris Anagnostou
>>>
>>> From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
>>> To: dimitris anagnostou <dimmechan at yahoo.com>
>>> Cc: Bobby Treat <drBob at bigfoot.com>; Dr. Wolfgang Hintze
>> <weh at snafu.de>; "mathgroup at smc.vnet.net Steve" 
> <mathgroup at smc.vnet.net>;
>>
>> Peter Pein <petsie at dordos.net>
>>> Sent: Thursday, October 6, 2011 9:48 PM
>>> Subject: Re: Re: simplification
>>>
>>>
>>> On 6 Oct 2011, at 20:28, Andrzej Kozlowski wrote:
>>>
>>>>
>>>> On 6 Oct 2011, at 16:57, Andrzej Kozlowski wrote:
>>>>
>>>>>
>>>>> On 5 Oct 2011, at 10:01, Peter Pein wrote:
>>>>>
>>>>>> Am 04.10.2011 07:40, schrieb dimitris:
>>>>>>> Hello.
>>>>>>>
>>>>>>> Let
>>>>>>>
>>>>>>> o1 = 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]];
>>>>>>> o2 = 1 + Sqrt[3] + Sqrt[5] + Sqrt[7];
>>>>>>>
>>>>>>> o1 is equal to o2.
>>>>>>>
>>>>>>> o1 == o2 // FullSimplify
>>>>>>> True
>>>>>>>
>>>>>>> The question is how to make Mathematica to simplify o1 to o2.
>>>>>>>
>>>>>>> Thanks
>>>>>>> Dimitris
>>>>>>>
>>>>>>
>>>>>> With a lot of luck:
>>>>>>
>>>>>> In[1]:= o1 = 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 +
>> Sqrt[35])]];
>>>>>> ext = Block[{x, poly = RootReduce[o1][[1]]},
>>>>>> Sqrt[Cases[Union @@ Divisors[Abs[CoefficientList[poly[x], x]]],
>>>>>> 1 | _?PrimeQ, 1]]]
>>>>>> o2 = ((Rest[#1] / First[#1]) . ext & )[
>>>>>>    FindIntegerNullVector[Prepend[ext, -o1]]]
>>>>>>
>>>>>> Out[3]= {1, Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[7], Sqrt[19],
>> Sqrt[31]}
>>>>>>
>>>>>> Out[4]= 1 + Sqrt[3] + Sqrt[5] + Sqrt[7]
>>>>>>
>>>>>> :-)
>>>>>>
>>>>>
>>>>> Neat, but from the Mathematical point of view the question was
>> posed "the wrong way round" in that o1 is mathematically "simpler" =
> than
>> 1 + Sqrt[3] + Sqrt[5] + Sqrt[7], since it is already expressed in =
> terms
>> of its minimal polynomial. Hence this is the "natural" or "easy" way =
=
> to
>> go:
>>>>>
>>>>> ToRadicals[RootReduce[1 + Sqrt[3] + Sqrt[5] + Sqrt[7]]]
>>>>>
>>>>> 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]]
>>>>>
>>>>> in other words, the algebraic "simplification" in this case is
>> exactly the opposite of, what might be called, the visual one.
>>>>>
>>>>>
>>>>> There is no natural or unique way to "decompose" algebraic =
numbers=
>
>> that are already reduced into sums etc, of "simpler" summands or =
> factors
>>
>> etc. Of course, if we already know an integer basis for an algebraic==

>
>> number field containing an algebraic number, than there are ways of
>> expressing it in terms of this basis - and this method is an example.
>>>>>
>>>>> Andrzej Kozlowski
>>>>>
>>>>>
>>>>>
>>>>>
>>>>
>>>> Let me correct myself here since I noticed that I did not write =
what
>>
>> I really meant ;-)
>>>>
>>>> What I mean was that the following is the simplest (algebraically)==

>
>> form of this algebraic number:
>>>>
>>>> In[80]:= RootReduce[1 + Sqrt[3] + Sqrt[5] + Sqrt[7]]
>>>>
>>>> Out[80]= Root[
>>>> 1024 + 3584 #1 + 640 #1^2 - 1984 #1^3 - 48 #1^4 + 304 #1^5 -
>>>> 32 #1^6 - 8 #1^7 + #1^8 &, 8]
>>>>
>>>> (the same as above, but, of course, without the ToRadicals).
>> Mathematically equivalent statement is:
>>>>
>>>> MinimalPolynomial[RootReduce[1+Sqrt[3]+Sqrt[5]+Sqrt[7]],x]
>>>> 1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8 x^7+x^8
>>>>
>>>> The only simplification one can really make algorithmically with
>> algebraic numbers is what RootReduce does - essentially finding the
>> minimal polynomial. Mathematica also has an algorithm for converting==

>
>> some root objects to radicals but this is usually does not give the
>> visually simplest form. There is no algorithm that will discover the==

>
>> simples such form in general (indeed, most algebraic numbers can't =
be=
>
>> expressed in radicals at all). On the other hand, one can find the
>> minimal polynomial of any algebraic number: e.g.
>>>>
>>>> In[82]:= MinimalPolynomial[1 + Sqrt[3] + Sqrt[5] + Sqrt[7], x]
>>>> Out[82]= 1024 + 3584*x + 640*x^2 - 1984*x^3 - 48*x^4 + 304*x^5 -==

>
>> 32*x^6 - 8*x^7 + x^8
>>>>
>>>> In[83]:=
>> MinimalPolynomial[1+Sqrt[15+2*Sqrt[35]+2*Sqrt[6*(6+Sqrt[35])]],x]
>>>> Out[83]= 1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8
>> x^7+x^8
>>>>
>>>> You get the same answer which is how Mathematica knowns that these==

>
>> numbers are really equal - it does not attempt to transform one into =
=
> the
>>
>> other. When I wrote that
>>>> 1+Sqrt[15+2*Sqrt[35]+2*Sqrt[6*(6+Sqrt[35])] was "algebraically
>> simpler" than 1+Sqrt[3]+Sqrt[5]+Sqrt[7]] I meant only that the former =
=
> is
>>
>> Mathematica's radical representation of Root[1024 + 3584 #1 + 640 =
#1^2 =
> -
>>
>> 1984 #1^3 - 48 #1^4 + 304 #1^5 - 32 #1^6 - 8 #1^7 + #1^8 &, 8] - =
which=
>
>> is indeed the simplest way (algebraically) to express this algebraic==

>
>> number, which, of course, has infinitely many other representations =
in=
>
>> terms of radicals (which Mathematica will not be able, in general, =
to=
>
>> convert into one another but will always be able to find their =
minimal=
>
>> polynomial).
>>>>
>>>> This is not a limitation of Mathematica - it is a limitation of =
the=
>
>> radical representation of algebraic numbers.
>>>>
>>>> Andrzej Kozlowski
>>>>
>>>>
>>>>
>>>
>>>
>>> Even the above was still not quite accurate. MinimalPolynomial of =
an=
>
>> algebraic number, still does not determine it - obviously
>>>
>>> MinimalPolynomial[
>>> Root[1024 + 3584 #1 + 640 #1^2 - 1984 #1^3 - 48 #1^4 + 304 #1^5 -
>>>   32 #1^6 - 8 #1^7 + #1^8 &, 8], x]
>>>
>>> 024 + 3584 x + 640 x^2 - 1984 x^3 - 48 x^4 + 304 x^5 -
>>> 32 x^6 - 8 x^7 + x^8
>>>
>>> MinimalPolynomial[Root[1024+3584 #1+640 #1^2-1984 #1^3-48 #1^4+304
>> #1^5-32 #1^6-8 #1^7+#1^8&,7],x]
>>> 1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8 x^7+x^8
>>>
>>> but of course these roots themselves are not the same. However, =
once=
>
>> you know the minimal polynomial of an algebraic number you only need =
=
> to
>> isolate it from the other roots to determine it uniquely - which is
>> something that Mathematica does automatically when it numbers the =
> roots.
>>
>> So, once you determine the minimal polynomials of two algebraic =
> numbers
>> we know that if the polynomials are different so are the numbers, if==

>
>> they are the same we use root isolation to distinguish or identify =
> them.
>>
>> In any case, the point it that no transformations are performed on
>> radicals when doing this sort of thing.
>>>
>>>
>>>
>>
>>
>
>




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