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A very interesting problem from here.

I got the algebra in less than a minute with all 6 solutions (integers). But was puzzled a bit that the graph only gives 4. As now I have waken up and tried it on my own, these are the findings.

Plot[(x^2 - 7 x + 11)^(x^2 - 13 x + 42), {x, 0, 8}, Frame -> True,  PlotRange -> {{0, 8}, {-0.25, 3}},  GridLines -> {Range[8], {-1, 0, 1}}, Mesh -> All] 

enter image description here

Obviously there is a whole region "missing". Then I tried to plot it as a discrete function.

DiscretePlot[(x^2 - 7 x + 11)^(x^2 - 13 x + 42), {x, 0, 8},  Frame -> True, PlotRange -> {{0, 8}, {-0.25, 3}},  GridLines -> {Range[8], {-1, 0, 1}}] 

enter image description here

I have tried to tweak Mesh, PlotPoints, ect.

I know what's going on mathematically in that region, but is there a better way to explore this function when we plot it, so that we don't miss special points, especially "important" points like integer numbers that are still valid?

In comparison

NSolve[(x^2 - 7 x + 11)^(x^2 - 13 x + 42) == 1, x, Reals]

{{x -> 2.}, {x -> 5.}, {x -> 6.}, {x -> 7.}}

Failed to give all solutions in this case while

Solve[(x^2 - 7 x + 11)^(x^2 - 13 x + 42) == 1, x, Reals]

{{x -> 2}, {x -> 3}, {x -> 4}, {x -> 5}, {x -> 6}, {x -> 7}}

does give all correct answers.

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    $\begingroup$ Plot[ReIm[(x^2-7 x+11)^(x^2-13 x+42)], {x, 2, 5},PlotRange->All] $\endgroup$ – Bill Jan 4 at 19:00

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