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I'm trying to learn more about this such as methods for finding solutions, what it's useful for, etc. but I am not sure what it is called to be searching for to begin with.

What is it called when we have an equation typically with division or powers and roots or whatnot which makes most of the range not whole numbers or not map neatly from whole number to whole numbers, and we are only interested in (finding) the solutions which are whole numbers?

Michael
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  • Diophantine equations are equations where only the integer solutions matter. Are you looking for positive solutions of Diophantine equations? – Patrick Gambill Jan 06 '23 at 05:08
  • @PatrickGambill I did see diophantine equations but I'm not sure they cover anything outside simple polynomial equations. For instance suppose I have (ax+b)^m/2^n and want to find whole number solutions? – Michael Jan 06 '23 at 05:25
  • I'm not sure beyond that. Maybe someone else knows? – Patrick Gambill Jan 06 '23 at 05:30
  • @PatrickGambill ok, thanks! If it's not a field of study yet maybe I've invented it (haha!) – Michael Jan 06 '23 at 05:41
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    For instance suppose I have (ax+b)^m/2^n and want to find whole number solutions? --- The title says you're interested in equations but (ax+b)^m/2^n is not an equation. Also, is this $(ax+b)^m$ divided by $2^n,$ or is this $(ax+b)^{m/2}$ raised to the $n$th power, or is this $(ax+b)$ raised to the $(m/2)^n$ power, or something else? – Dave L. Renfro Jan 06 '23 at 07:50
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    @DaveL.Renfro Order of operations both as taught in textbooks and as used in practice when typing mathematics inline agree that that expression would be $\dfrac{(ax+b)^m}{2^n}$. – Mark S. Jan 06 '23 at 11:15
  • @Mark S.: used in practice when typing mathematics inline --- Assuming the writer knows this . . . – Dave L. Renfro Jan 06 '23 at 21:29

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As mentioned in the comments, when you're looking for integer solutions to an equation, you are studying Diophantine equations. They need not be polynomial equations, and you could restrict yourself to only caring about nonnegative/positive solutions.

For example, "Find when $\dfrac{(ax+b)^m}{2^n}$ is an integer" is basically the same as "find integer solutions to $(ax+b)^m=2^nc$", which is an exponential Diophantine equation.

Mark S.
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