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Does the formulation of a proof involve substitution of numerical solutions into the problem ?

I asked this because I was trying to find a way I can prove Erdos right in Brocard's problem using Diophantine equations (something I am not acquainted with except problems involving a system of linear Diophantine equations). My approach is like this : if we assume that the next solution $(m,n)$ is $(71 + x, 7 + y)$ do the following to get a new equation: $$(7 + y)! + 1 = (71 + x)^2 \longrightarrow(1)$$ $$7! + 1 = 71^2 \longrightarrow(2)$$ $$(1) - (2) = (7 + y)! - 5040 = x^2 + 142x\longrightarrow(3)$$ And then prove that only a finite number of integer solutions exist for (3) and out of them, only 3 satisfy the constraints in Brocard's problem. I doubt whether what I am doing is proof-writing or not. I got a few numerical solutions using Wolfram Alpha and those are what I am using.

Edit : The rest of the solutions given by Wolfram Alpha also worked, except that $71 + x$ became negative in those solutions I rejected by accident.

Spectre
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    Well, the base case of an induction proof involves substitution of a small number. The problem with your proposal is that it can only rule out solutions that fit your scheme, it's oblivious to other possible solutions that don't conform to that pattern. That's not fatal, though. Eg, the standard proof for perfect numbers from Mersenne primes only works for even perfect numbers. There might be odd perfect numbers, but they're invisible to that proof. – PM 2Ring Jan 20 '21 at 02:02
  • @PM2Ring , so basically, what I am suggesting is usable but not a very right way of proving ? – Spectre Jan 20 '21 at 02:04
  • @PM2Ring if you check in Wolfram Alpha, you can see that it shows up some integer solutions for equation (3) , out of which 3 proved to be solutions to (1) as well. I am not sure about the rest (since I didn't try them). – Spectre Jan 20 '21 at 02:12
  • Even Dr James Grime (that young mathematician who appears in Numberphile videos) says that it is hard to make a proof. I asked him via email. – Spectre Jan 20 '21 at 02:13
  • I believe Wolfram Alpha is good at solving non-linear Diophantine equations with factorials and a whole lot of that. – Spectre Jan 20 '21 at 02:14
  • Oh well, sorry; the rest of the solutions to (3) also seem to be applicable to (1) ':) I just checked them now – Spectre Jan 20 '21 at 02:18
  • And they are the only solutions shown by Wolfram Alpha... – Spectre Jan 20 '21 at 02:21
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    On a related note, see https://math.stackexchange.com/q/111440/207316 – PM 2Ring Jan 20 '21 at 02:59
  • "Wolfram Alpha is great, but it's not perfect. I can't recall the details right now, but a month or so ago I tried to use it to see if there were additional solutions to something I was working on. But it didn't even find the solution that I'd found! " - that makes sense.. so is there anything I can rely on ? – Spectre Jan 20 '21 at 03:01
  • I am 15 years old, so I may not be able to solve complex problems like the advanced mathematicians I see in here. – Spectre Jan 20 '21 at 03:02
  • Let us continue this discussion in chat. – PM 2Ring Jan 20 '21 at 03:06

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