Hi,
I need to help with solution of diofantine equations 4x+2y=28 in Integers.
If I find Documentation Center, it say: Reduce[4x+2y==28,{x,y},Integers].
But the result is not expressed with the parameter C like in the picture.
THANKS A LOT!
Are you interested in the linear or quadratic equation? Also, what would be a form of result that you are trying to obtain?
She searches for integer solutions and does not believe that the expressions given are integer. Mathematica is also not sure about that:
In[51]:= Clear[x, y]
x[n_] := 1/2 ((8 - 3 Sqrt[7])^n + (8 + 3 Sqrt[7])^n)
y[n_] := -(((8 - 3 Sqrt[7])^n - (8 + 3 Sqrt[7])^n)/(2 Sqrt[7]))
In[68]:= RootApproximant[x[#]] & /@ RandomInteger[{1, 30}, 17]
Out[68]= {2024, 2199950293420883836928, 2024, \
2325668404237133588966684445638656, 2261936092886171715528294400, \
2261936092886171715528294400, 8, 2261936092886171715528294400,
1/2 (134255446617603 + Sqrt[
18024524946491071433658089473]), 574522862194843517270624305152, \
2024, 8193151, 2199950293420883836928, 558778713173023503941632, 8, \
37064767922732740871188780993740800, \
2325668404237133588966684445638656}
In[69]:= RootApproximant[y[#]] & /@ RandomInteger[{1, 30}, 17]
Out[69]= {12535521795, 194307, 765, 13625260145284696589750763520, \
13625260145284696589750763520, 3096720, 854931483328272233719136256, \
12192, 765, 831503053299319046144, 3183973182717, \
3460762433314345591306287316992, 786554688, \
854931483328272233719136256, 3, 3365924154344798473420800, 48}
and going up to 30 is not so far.
In[72]:= FactorInteger[18024524946491071433658089473]
Out[72]= {{151, 1}, {2577984271, 1}, {46302732029905513, 1}}
In[73]:= 151 2577984271 == 46302732029905513
Out[73]= False
Indeed not a square of something.
By the way, Martina, the solution appears exactly as announced by the manual:
In[48]:= Reduce[x^2 - 7 y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]
Out[48]= C[1] \[Element] Integers && C[1] >= 1 &&
x == 1/2 ((8 - 3 Sqrt[7])^C[1] + (8 + 3 Sqrt[7])^C[1]) &&
y == -(((8 - 3 Sqrt[7])^C[1] - (8 + 3 Sqrt[7])^C[1])/(2 Sqrt[7]))
The linear equation has of course also a constant, but this is pointless,isn’t it?
In[74]:= Reduce[4 x + 2 y == 28, {x, y}, Integers]
Out[74]= C[1] \[Element] Integers && x == C[1] && y == 14 - 2 C[1]
You can try another function:
Solve[4 x + 2 y == 28, {x, y}, Integers]
and receive
{{x → ConditionalExpression[C[1], C[1] [Element] Integers],
y → ConditionalExpression[14 - 2 C[1], C[1] [Element] Integers]}}