**Diophantine equations are indeterminate polynomial equations in which only
integer solutions are allowed.** (1)For a 1st order Diophantine equation,
there is an algorithm to determine whether there exists a solution. A linear
Diophantine equation of 2 variables is of the form Ax + By = C where A, B and C
are integers. The first known solution was furnished by Brahmagupta.
(2)Fibonacci numbers have a diophantine representation. A
pair of consecutive fibonacci numbers do indeed satisfy the equation: **
x**^{2}+xy-y^{2} = ±1 and
the solutions (x, y) in positive integers to the equation **
(x**^{2}+xy-y^{2})^{2} - 1= 0
are the pairs of consecutive Fibonacci numbers. (3) The
linear Diophantine equation **ax-by=c** with integers a, b, c is solvable
if and only if gcd of a and b divides c OR ** (b, a) |c. Example- **take 38x-15y=1. Various solutions (x, y)
are (-13,-33),(2,5),(17,43). solutions yield good approximations to 15/38 i.e.
(x/y) --> 15/38.One can work out on other equn. 3x-5y=19, 117x-125y=91
etc. If the linear Diophantine equation is solvable, there are infinite
number of integer solutions. (4) Euclid showed that if
there exist 2 integers a, b , then gcd (a, b) can be expressed as a linear
combination of a and b. Or ax + by = gcd (a, b) |

**Some Observations**
**(1)** If Ax -By=C is a
Diophantine equation, and a solution is found out at x=m, y=n,
then (m+ B, n+ A),(m+2B,n+2A),(m+3B,n+3A,...(m+ kB, n+ kA) are
also solutions where k is an integer (positive or negative).
**(2)** If Ax -By=C is a
diophantine equation & the lowest difference between x, y i.e. x-y =Δ
, then in general x-y = Δ +k (B-A) where k is an integer.
**(3) **If Ax -By=C is a
Diophantine equation & the lowest sum between x, y i.e. x+ y =δ
, then in general x+ y =δ + k (B+ A) where k is an integer, positive or
negative.
**(4) **If Ax -By = C is a
Diophantine equation, no. of lowest values that y can take is
**A** irrespective
of the value of C and B. For example, if 2x-5y=C, the lowest value of y can
be 1 or 2--> only these 2 values whether C is 1,2,3,4,17 etc and /or B is
of
any value. Again , if 3x-5y=C, lowest y can take only any of the 3 values (1,2 or
3). If 6x-5y=C, y shall take any of 6 values (1,2,3,4,5,6) irrespective of
value of B, C.
**(5) **If Ax -By = C is a Diophantine
equation where y= m, then y = m for the equation Ax -By = C ± kA
where k is an integer. Example -- In the equation, 15x-7y =4 , y=8 (lowest
value). Then y = 8 (lowest value) also for the equation 15x-7y =19 and
15x-7y=34 etc.
**(6) **If Ax -By = C is a Diophantine
equation for which lowest value of y is m, then for Ax-(B+kA)y = C, y=m
where k is an integer.
**(7) **If Ax-By = C is a diophantine
equation & lowest value of y is m,
then for kAx-kBy=kC, lowest value of y shall also be m
where k is an integer.
**(8) **If Ax-By = C is a diophantine
equation & lowest value of y is m
and Ax-B1y = C,Ax-B2y = C,Ax-B3y = C etc upto Ax-(A-1)y = C are various
diophantine equations where B=1,B1=2,B2=3 etc and lowest value of y are
m1,m2,m3,m4 etc , then for a specific value of Bm, if there exist any
combination of products with same value,then these are also values of some B
and m provided m<=A.
Example:-39x- y =3 where lowest value of y i.e m=36. So here Bm= 36. Other
combination with product 36 are (18,2) ,
(9,4),(6,6),(4,9),(2,18),(36,1).Hence other diophantine equations are
**39x-18y= 3** Lowest value of y= 2 (18,2)
**39x-9y = 3** Lowest value of y= 4 (9,4)
**39x-6y= 3** Lowest value of y = 6 (6,6)
**39x-4y= 3** Lowest value of y = 9 (4,9)
**39x-2y= 3** Lowest value of y = 18 (6,6)
**39x-36y= 3** Lowest value of y = 1 (36,1)
Here all Bm are 36.
In the same way if 39x-5y=3 & lowest value of y is 15 then Bm is 75
(5,15).Other combinations are (3,25),(25,3),(15,5). Corresponding
diophantine equations are
**39x-3y= 3** value of y= 25 (3,25)
**39x-25y= 3** value of y= 3 (25,3)
**39x-15y= 3** value of y= 5 (15,5)
All are acceptable values of y as these are <=39. (75,1) is also an option
but not acceptable as B=75 which is greater than 39.
**(9) **In the above case where A,C remain
constant and B changes leading to various values of y,the various values of
Bm are lowest value of **Bm +kA** where k is
an integer.In example above, since lowest value of Bm is 36, other values
are 75,114,153,192,231,270,309 .....1050.(Some values are excluded such as
543 as it is (3,181) and (181,3) out of which 181 value of y is not the
lowest value and 181 value of B is greater than A. However, they correspond
to a pair of values of B,y which form diophantine equations) One can take their product
combinations and construct diophantine equations with value of y. Example--
If we take 270, the combinations are
(135,2),(2,135)(45,6),(6,45),(15,18)(18.15),27,10), (10,27).Excluding
the red colored combinations which are not lowest value of y, other form the
equation such as 39x-15y=3 where y=18 and so on so forth. |