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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
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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

Thank you very much for the effort. Using Pari/GP I have the full set of (even) solutions up to $x \approx 100$ Interesting, I thought there are only even solutions... But well, my concern was -

  • 1) to get some analytical answer on bounds etc hoping to understand the underlying mechanism (I want to generalize to 4 and more unknowns and see what one can get from this approaching the "low-level"-answers to Lehmer's totient problem for the small cases (thus only even solutions were relevant to me, but to see the odd solutions as well this might well be helpful for the understanding of the principles)

  • 2) to make the W|A team aware that their software says "does not exist" instead of "didn't analyze deep enough, try with more space/time" or so.

Currently I look at the bounds for $y,z$ when $y=z$ and $x$ is given. While it seems an absolute bound is always $x,y=(1+x)^2-1,z=(1+x)^3-1$ we have cases where $x,y>(1+x)^2-1,z>y$ but there seems to be a limit to which $y$ and $z$ converge towards each other when $x$ is increased (it is not something like arithmetic or geometric mean when some $x$ is taken, I do not have the exact function on $x$ yet, only a pair of curves which indicate that indeed the trends of the $y$ and of the $z$ do not overlap and is always "separable").
Just looking at your odd cases - for instance $(x,y,z)=(3,15,63)$ at least they also follow that "maximal interval" of showing $(x+1,y+1,z+1)=4,4^2,4^3$ . I have at least an argument why this does no more work for higher powers of $(x+1)$ but not yet, why not larger intervals are possible at all - for analytical reasons. So the loop to find the complete set of solutions for the searchintervals in $x,y,z$ is $x=2,.. arbitrary; y=x..(1+x)^{2+c}, z=y..(1+x)^3-1)$ where the exponent $2+c=f(x+1)$ for the upper bound of $y$ is subject of my current analysis (of course here only solutions are sought with $x \le y \le z$ )
POSTED BY: Gottfried Helms
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