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Let's say that you want to prove a statement is true for all prime numbers. Can this be done by Math Induction?

Zapper Nona
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    Sure. First prove that if it is true for some prime, it is true then for the next one. Then prove it is true for the first prime and you're done. – K.defaoite May 31 '20 at 13:39
  • So you can use Math Induction to prove a statement is true for all prime numbers because they are all positive, right? – Zapper Nona May 31 '20 at 13:41
  • Try it for Cauchy's theorem. If a prime $p$ divides the order of $G$, then $G$ has an element of order $p$. This is true for all primes. – Dietrich Burde May 31 '20 at 13:41
  • It can be done with Cauchy's Theorem, but mathematically speaking, it can be done by only using Math Induction, correct? – Zapper Nona May 31 '20 at 13:42
  • Cauchy's Theorem can be proved by strong induction, yes, but it is not so straightforward. – Dietrich Burde May 31 '20 at 13:42
  • @Zapper Nona He means proving Cauchy's theorem by induction. – K.defaoite May 31 '20 at 13:42
  • Ok I am a little confused now, sorry. I don't want to relate Cauchy's Theorem or anything - just simple Math Induction. Let's say we have a statement, and want to prove if it is true for all prime numbers - could this be, 1. solved by Math Induction because the statement regards all positive numbers, 2. Can't solve because it doesn't relate all positive integers, 3. because numbers aren't evenly spaced, or 4. you can use math induction if you ignore 2. – Zapper Nona May 31 '20 at 13:45
  • Well, think about FLT. Can it be done? – Dietrich Burde May 31 '20 at 13:46
  • So would it be 1, 2, 3, or 4? (look at my comment above). Thanks! – Zapper Nona May 31 '20 at 13:48
  • Fermat concerns all positive integers and all prime numbers. – Dietrich Burde May 31 '20 at 13:49
  • So would the answer be 1, 2, 3, or 4? (for math induction only, not FLT) – Zapper Nona May 31 '20 at 13:49
  • No, none of 1,2,3,4 for FLT by induction. – Dietrich Burde May 31 '20 at 13:50
  • Wait actually but I am asking for Math Induction only, not FLT - i know it is not known if it can be solved by Math Induction, but Math Induction only, not FLT, would it be 1, 2, 3, or 4? – Zapper Nona May 31 '20 at 13:51
  • I don't understand. Your question asks for a "statement" (so FLT here), to be proved by induction over primes. So this is unknown for "this statement". – Dietrich Burde May 31 '20 at 13:53
  • I do not want to use the FLT here. For any particular statement, I want to know if it could be proved by Math Induction only - for all prime numbers (i know it is not known, but out of my options 1,2,3,4 above), which one would work best? – Zapper Nona May 31 '20 at 13:54
  • "For any particular statement" you cannot exclude a negative result, like for FLT. You want to exclude FLT - good. But then you have to exclude many other statements, so this doesn't make sense. – Dietrich Burde May 31 '20 at 13:55
  • Alright I understand. Thank you for your help! – Zapper Nona May 31 '20 at 13:57
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    You need to be precise about what you mean by "mathematical induction." It's perfectly correct to prove "For all primes $p$, property $A(p)$ holds" by showing (1) $A(2)$ holds, and (2) For any prime $p$, if $A(p)$ holds, then $A(p)$ holds [where $p$ denotes the next prime after $p$]. However unlike more basic induction arguments, you don't have a simple fixed formula for $p*$ in terms of $p$. But the logic is fine, if that's your question. – Ned May 31 '20 at 14:35
  • That was my question! Thank you! – Zapper Nona May 31 '20 at 14:44

1 Answers1

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Let's say that you want to prove a statement is true for all prime numbers.
Such as Fermat's Last Theorem:

$$ X^p+Y^p=Z^p$$

has no non-trivial integer solutions for all primes $p>2$.

Can this be done by Math Induction?

This is not known.

Dietrich Burde
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