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From: riderofgiraffes on 10 Jun 2010 11:56 This isn't new, this isn't deep, and it isn't a mystery, but I sometimes use this to get kids interested in proof. For prime p>3, p^2-1 is a multiple of 24. It's not hard to prove, but it's a good one to get kids thinking about how we can know something is always true without just checking every number. Just an idle thought. I'll now return you to your usual program of deep maths and troll baiting.
From: José Carlos Santos on 10 Jun 2010 17:07 On 10-06-2010 20:56, riderofgiraffes wrote: > This isn't new, It was for me. > this isn't deep, and it isn't a mystery, > but I sometimes use this to get kids interested in proof. > > For prime p>3, p^2-1 is a multiple of 24. > > It's not hard to prove, but it's a good one to get kids > thinking about how we can know something is always true > without just checking every number. Very good example. Best regards, Jose Carlos Santos
From: TCL on 10 Jun 2010 19:15 On Jun 10, 3:56 pm, riderofgiraffes <mathforum.org...(a)solipsys.co.uk> wrote: > This isn't new, this isn't deep, and it isn't a mystery, > but I sometimes use this to get kids interested in proof. > > For prime p>3, p^2-1 is a multiple of 24. > > It's not hard to prove, but it's a good one to get kids > thinking about how we can know something is always true > without just checking every number. > > Just an idle thought. I'll now return you to your usual > program of deep maths and troll baiting. Here is an improvement: For any odd number n not divisible by 3, n^2-1 is a multiple of 24. -TCL
From: Liviu on 11 Jun 2010 00:43 "TCL" <tlim1(a)cox.net> wrote... > On Jun 10, 3:56 pm, riderofgiraffes wrote: >> >> This isn't new, this isn't deep, and it isn't a mystery, >> but I sometimes use this to get kids interested in proof. >> >> For prime p>3, p^2-1 is a multiple of 24. >> >> It's not hard to prove, but it's a good one to get kids >> thinking about how we can know something is always true >> without just checking every number. > > Here is an improvement: > > For any odd number n not divisible by 3, n^2-1 is a multiple of 24. Right, of course. However, whether that's an improvement or not (pedagogically speaking) largely depends on the target audience and intended point. The original statement leaves room for "relevancy of the hypothesis" followups like "now that you've proved it, where was the primeness of p used in the proof? ...or, if it wasn't, then what could be a more general condition on p for the conclusion to hold". Liviu
From: alainverghote on 11 Jun 2010 04:25
On 11 juin, 06:43, "Liviu" <lab...(a)gmail.c0m> wrote: > "TCL" <tl...(a)cox.net> wrote... > > On Jun 10, 3:56 pm, riderofgiraffes wrote: > > >> This isn't new, this isn't deep, and it isn't a mystery, > >> but I sometimes use this to get kids interested in proof. > > >> For prime p>3, p^2-1 is a multiple of 24. > > >> It's not hard to prove, but it's a good one to get kids > >> thinking about how we can know something is always true > >> without just checking every number. > > > Here is an improvement: > > > For any odd number n not divisible by 3, n^2-1 is a multiple of 24. > > Right, of course. However, whether that's an improvement or not > (pedagogically speaking) largely depends on the target audience and > intended point. The original statement leaves room for "relevancy of > the hypothesis" followups like "now that you've proved it, where was > the primeness of p used in the proof? ...or, if it wasn't, then what > could be a more general condition on p for the conclusion to hold". > > Liviu Good morning, We may also consider the product P = (n-1)n(n+1)(n+2) always a multiple of 24, more,since (n-1)(n+2)= n^2+n-2, n(n+1)= n^2+n P+1 is a square (n^2+n-1)^2 Alain |