Message Boards Message Boards

0
|
3806 Views
|
2 Replies
|
2 Total Likes
View groups...
Share
Share this post:

existing solutions on system of three modular equations not found by W|A.

Gottfried Helms, Univ of Kassel
Posted 5 years ago

I'm studying the system of 3 modular equations on $(x,y,z) \ge 1$:

$$ x + y + xy \equiv z \tag {eqnsys 1} \\ y + z + yz \equiv x \\ z + x + zx \equiv y $$
I've already found, that all (x,y,z) must be even, and heuristically it seems, that for $X=1+x,Y=1+y,Z=1+z$ we have always the bounds depending on $x$ as the smallest value for $Y=X^2, Z=X^3$ and that this $Z$ is unconditionally the largest when keeping $X$ and changing $Y$. I'm looking for a proof of that. and asked W|A for a possibly better analytical answer. That was not possible. So first I set $x=8$ to reduce the search space. W|A says, no solutions. I know that $(x,y,z)=(8,2,2),(8,2,26)$ are solutions. However feeding also $y=2$ W|A found $(z)=(8,2,2),(8,2,26)$.
Is it possible to get an algebraical expression for the set of solutions when $y$ (or even $x$ and $y$) are kept as unknown?
W|A- Code(providing $y$ the value $8$)= solve x+8+8x=0 (mod z),8+z+8z=0(mod x),z+x+xz =0(mod 8)

Update: the mathematical portion of my post seems to be solved, see my own answer in MSE: link to my question and answer in MSE

POSTED BY: Gottfried Helms
2 Replies
Sort By:
Bill Nelson
Bill Nelson
Posted 5 years ago

If I have understood your question correctly then I have been unable to get W|A to provide solutions, but these Mathematica solutions solutions might be of use to you.

Reap[For[x=2,x<=64,x+=2,For[y=2,y<=64,y+=2,For[z=2,z<=64,z+=2,
  If[Mod[x+y+x y,z]==Mod[y+z+y z,x]==Mod[z+x+x z,y]==0,Sow[{x,y,z}]]]]]][[2,1]]

which returns

{{2, 2, 2}, {2, 2, 4}, {2, 2, 8}, {2, 4, 2}, {2, 4, 14}, {2, 8, 2}, {2, 8, 26},
 {2, 10, 16}, {2, 14, 4}, {2, 16, 10}, {2, 26, 8}, {4, 2, 2}, {4, 2,14}, {4, 4, 4},
 {4, 4, 8}, {4, 4, 12}, {4, 4, 24}, {4, 6, 34}, {4, 8, 4}, {4, 8, 44},{4, 12, 4},
 {4, 12, 16}, {4, 12, 64}, {4, 14, 2}, {4, 16, 12}, {4, 16, 28}, {4, 24, 4},
 {4, 28, 16}, {4, 34, 6}, {4, 44, 8}, {4, 64, 12}, {6, 4, 34}, {6, 6, 6},
 {6, 6, 12}, {6, 6, 24}, {6, 6, 48}, {6, 12, 6}, {6, 12, 18}, {6, 12, 30},
 {6, 18, 12}, {6, 22, 40}, {6, 24, 6}, {6, 30, 12}, {6, 34, 4}, {6, 40, 22},
 {6, 48, 6}, {8, 2, 2}, {8, 2, 26}, {8, 4, 4}, {8, 4, 44}, {8, 8, 8}, {8, 8, 16},
 {8, 8, 40}, {8, 16, 8}, {8, 26, 2}, {8, 28, 52}, {8, 40, 8}, {8, 44, 4},
 {8, 52, 28}, {10, 2, 16}, {10, 10, 10}, {10, 10, 20}, {10, 10, 30},{10, 10, 40},
 {10, 10, 60}, {10, 16, 2}, {10, 20, 10}, {10, 30, 10}, {10, 40, 10},{10, 40, 50},
 {10, 50, 40}, {10, 60, 10}, {12, 4, 4}, {12, 4, 16}, {12, 4, 64}, {12, 6, 6},
 {12, 6, 18}, {12, 6, 30}, {12, 12, 12}, {12, 12, 24}, {12, 16, 4}, {12, 18, 6},
 {12, 24, 12}, {12, 24, 36}, {12, 30, 6}, {12, 36, 24}, {12, 36, 60},{12, 60, 36},
 {12, 64, 4}, {14, 2, 4}, {14, 4, 2}, {14, 14, 14}, {14, 14, 28}, {14, 14, 56},
 {14, 28, 14}, {14, 56, 14}, {16, 2, 10}, {16, 4, 12}, {16, 4, 28}, {16, 8, 8},
 {16, 10, 2}, {16, 12, 4}, {16, 16, 16}, {16, 16, 32}, {16, 16, 48}, {16, 28, 4},
 {16, 32, 16}, {16, 48, 16}, {16, 48, 64}, {16, 64, 48}, {18, 6, 12}, {18, 12, 6},
 {18, 18, 18}, {18, 18, 36}, {18, 36, 18}, {18, 36, 54}, {18, 54, 36},{20, 10, 10},
 {20, 20, 20}, {20, 20, 40}, {20, 40, 20}, {22, 6, 40}, {22, 22, 22},{22, 22, 44},
 {22, 40, 6}, {22, 44, 22}, {24, 4, 4}, {24, 6, 6}, {24, 12, 12}, {24, 12, 36},
 {24, 24, 24}, {24, 24, 48}, {24, 36, 12}, {24, 48, 24}, {26, 2, 8}, {26, 8, 2},
 {26, 26, 26}, {26, 26, 52}, {26, 52, 26}, {28, 4, 16}, {28, 8, 52},{28, 14, 14},
 {28, 16, 4}, {28, 28, 28}, {28, 28, 56}, {28, 52, 8}, {28, 56, 28}, {30, 6, 12},
 {30, 10, 10}, {30, 12, 6}, {30, 30, 30}, {30, 30, 60}, {30, 60, 30},{32, 16, 16},
 {32, 32, 32}, {32, 32, 64}, {32, 64, 32}, {34, 4, 6}, {34, 6, 4}, {34, 34, 34},
 {36, 12, 24}, {36, 12, 60}, {36, 18, 18}, {36, 18, 54}, {36, 24, 12},{36, 36, 36},
 {36, 54, 18}, {36, 60, 12}, {38, 38, 38}, {40, 6, 22}, {40, 8, 8}, {40, 10, 10},
 {40, 10, 50}, {40, 20, 20}, {40, 22, 6}, {40, 40, 40}, {40, 50, 10},{42, 42, 42},
 {44, 4, 8}, {44, 8, 4}, {44, 22, 22}, {44, 44, 44}, {46, 46, 46}, {48, 6, 6},
 {48, 16, 16}, {48, 16, 64}, {48, 24, 24}, {48, 48, 48}, {48, 64, 16},{50, 10, 40},
 {50, 40, 10}, {50, 50, 50}, {52, 8, 28}, {52, 26, 26}, {52, 28, 8},{52, 52, 52},
 {54, 18, 36}, {54, 36, 18}, {54, 54, 54}, {56, 14, 14}, {56, 28, 28},{56, 56, 56},
 {58, 58, 58}, {60, 10, 10}, {60, 12, 36}, {60, 30, 30}, {60, 36, 12},{60, 60, 60},
 {62, 62, 62}, {64, 4, 12}, {64, 12, 4}, {64, 16, 48}, {64, 32, 32},{64, 48, 16},
 {64, 64, 64}}

There appear to be some other solutions which are not limited to being even.

{{3, 3, 3}, {3, 3, 15}, {3, 15, 3}, {3, 15, 63}, {3, 63, 15},{5, 5, 5}, {5, 5, 35},
 {5, 35, 5}, {7, 7, 7}, {7, 7, 21}, {7, 7, 63}, {7, 21, 7}, {7, 63, 7},{9, 9, 9},
 {11, 11, 11},{13, 13, 13}, {13, 13, 39}, {13, 39, 13},{15, 3, 3}, {15, 3, 63},
 {15, 15, 15}, {15, 63, 3},{17, 17, 17}, {19, 19, 19}, {19, 19, 57}, {19, 57, 19},
 {21, 7, 7}, {21, 21, 21},{23, 23, 23},{25, 25, 25}, {27, 27, 27},{29, 29, 29},
 {31, 31, 31},{33, 33, 33},{35, 5, 5}, {35, 35, 35},{37, 37, 37}, {39, 13, 13},
 {39, 39, 39},{41, 41, 41},{43, 43, 43},{45, 45, 45}, {47, 47, 47},{49, 49, 49},
 {51, 51, 51},{53, 53, 53}, {55, 55, 55},{57, 19, 19}, {57, 57, 57}, {59, 59, 59},
{61, 61, 61}, {63, 3, 15}, {63, 7, 7}, {63, 15, 3}, {63, 63, 63}}

If you need other solutions and can provide constraints then I would take a few moments to see if I can find any more for you.
POSTED BY: Bill Nelson
POSTED BY: Gottfried Helms
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Tag limit exceeded
Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract