SMARANDACHE NUMBER RELATED TRIANGLES


                         K. R. S. Sastry
     Jeevan Sandhya, Doddakalsandra Post, Raghuvana Halli
                    Bangalore 560 062, INDIA



                             ABSTRACT

     Given a triangle in Euclidean geometry it is well known that
there exist an infinity of triangles each of which is similar to
the given one.  In Section I we make certain observations on
Smarandache numbers.  This enables us to impose a constraint on
the lenghts of the corresponding sides of similar triangles.  In
Section II we do this to see that infinite class of similar
triangles reduces to a finite one.  In Section III we disregard
the similarity requirement.  Finally, in Section IV we pose a set
of open problems.

     I SMARANDACHE NUMBERS: SOME OBSERVATIONS

     Suppose a natural number n is given.  The Smarandache number
of n is the least number denoted by S(n) whith the following
property: n divides S(n)! but not (S(n)-1)!.  Below is a short
table containing n and S(n) for 1 ó n ó 12.

  n    1  2  3  4  5  6  7  8  9  10  11 12 
  S(n) 1  2  3  4  5  3  7  4  6   5  11  4
     
     A look at the above table shows that S(3) = S(6) = 3, 
S(4) = S(8) = S(12) = 4, ... .
     Let a natural number k be given.  Then the equation S(x) = k
cannot have an infinity of solutions x.  This is because the
largest solution is x=k!.  This observation enables us to impose
a restriction on the lengths of the corresponding sides of
similar triangles.  In the next section we shall see how to do
this.  Throughout this paper the triangles are assumed to have
natural number side lengths.  Also, the triangles are non
degenerate.

     II SMARANDACHE SIMILAR TRIANGLES

     Let us denote by T(a,b,c) the triangle ABC with side lengths
a, b, c.  Then the two similar triangles T(a,b,c) and
T'(a',b',c') are said to be Smarandache Similar if S(a) = S(a'),
S(b) = S(b'), S(c) = S(c').  Trivially, a given triangle is
Smarandache similar to itself.  Non trivially the two Pytagorean
triangles (right triangles with natural number side lengths)
T(3,4,5) and T'(6,8,10) are Smarandache similar because 
S(3) = S(6) = 3, S(4) = S(8) = 4, S(5) = S(10) = 5.  
However, the Similar triangles (3,4,5) and (9,12,15) are not
Smarandache Similar because S(3) = 3 different from S(9) = 6.  In 
fact the class of Smarandache Similar triangles generated by 
T(3,4,5) contains just two: T(3,4,5) and T'(6,8,10) in view of the 
fact that the solution set of the equation S(x) = 3 consists of 
just two members x = 3, 6.
     For another illustration let us determine the class of
Smarandache Similar Triangles generated by the 60 degrees triangle
T(a,b,c) = (5,7,8).  The algorithm to do this is as follows:
First we calculate S(5) = 5, S(7) = 7, S(8) = 4.  Next we solve
the equations S(a') = 5, S(b') = 7, S(c') = 4.  Let us solve the
last equation first.
     S(c') = 4 --> c' = 4, 8, 12, 24.
Here the largest value c' = 24 = 3c.  Hence we need not to solve
the other two equations beyond the solutions a' = 3a, b' = 3b. 
This observation therefore gives us
     S(a') = 5 --> a' = 5, 10, 15   and
     S(b') = 7 --> b' = 7, 14, 21.
It is now clear that the class of Smarandache similar triangles
contains just two members: (5,7,8) and (15,21,24).


     III SMARANDACHE RELATED TRIANGLES

     In this section we do not insist on the similarirty
requirement that we had in Section II.  Hence the definition:
Given a triangle T(a,b,c) we say that a triangle T'(a',b',c') is
Smarandache related to T if S(a') = S(a), S(b') = S(b), S(c') =
S(c).  Note that the triangles T and T' may or may not be
similar.  As an illustration let us determine all the triangles
that are Smarandache related to T(3,4,5).  To do this we follow
the same algorithm that we mentioned in Section II but we have to
find all the solutions of the equations S(a') = 3, S(b') = 4, 
S(c') = 5. Therefore
     S(a') = 3 --> a' = 3, 6;
     S(b') = 4 --> b' = 4, 8, 12, 24;
     S(c') = 5 --> c' = 5, 10, 15, 20, 30, 40, 60, 120.
This gives us the complete solution (a',b',c') = (3,4,5);
(3,8,10); (3,12,10); (6,4,5); (6,8,5); (6,8,10); (6,12,10);
(6,12,15); (6,24,20).

               IV CONCLUSION

     In the present discussion I have used small natural numbers
k so that the solution of the  equations S(x) = k can be easily
determined.  I do not know if this interesting converse problem
of determining all natural numbers x for given k of the
Smarandache equation S(x)=k has been disscused by someone
already.  In case if this has not been already considered, I
invite the reader to devise efficient methods to solve the
preceding equation.  We conclude this section by posing the
following open problems to the reader.
     (A)  Are there two distinct dissimilar Phytagorean triangles
that are Smarandache related?  i.e. both T(a,b,c) and
T'(a',b',c') are Pythagorean such that S(a') = S(a), S(b') =
S(b), S(c') = S(c) but T and T' are not similar.
     (B)  Are there two distinct and dissimilar 60 degrees 
triangles (120 degrees triangles) that are Smarandache related?
     (c)  Given a triangle T(a,b,c).  Is it possible to give
either an exact formula or an uper bound for the total number of
triangles (without actually determining all of them) that are
Smarandache related to T?
     (D)  Consider other ways of relating two triangles in the
Smarandache number sense.  For example, are there two triplets of
natural numbers (a,b,c) and (a',b',c') such that a + b + c = a' +
b' + c' = 180 and S(a) = S(a'), S(b) = S(b'), S(c) = S(c').  
If such distinct triplets exist the two triangles would be
Smarandache related via their angles.  Of course in this
relationship the side lengths of the triangles may not be natural
numbers.