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Cool Maths thing – square numbers

June 7, 2012

Just spotted on twitter that @cuttheknotmath and @algebrafact had tweeted the following fact. For integer p,
p(p + 1)(p + 2)(p + 3) + 1
Is a perfect square. This is cool in itself. Then @MrP_tchr tweeted that a colleague had spotted if you multiply the biggest and smallest numbers together and add one, you get the square root. ie
p(p + 3) + 1

And

(p + 2)(p + 3) – 1

Is also a square root. I believe that seeing things like this and thinking “wow, that’s cool” is pretty much the definition of being a mathematician.

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2 Comments
  1. Timon Piccini permalink

    I had a student solve a perfect square with this thought.Me: "What is 14 squared?"S: "Well 13^2 is 169, and so 13*14 is 182, so 14^2 is 196!"I thought it was awesome, and I had never that n^2= (n-1)^2+(n-1)+(n)I thought that was clever.

  2. MoreThanMaths permalink

    I love these sorts of results. Couldn???t resist having a play around with this. p(p + 1)(p + 2)(p + 3) + 1 = p??? + 6p?? + 11p?? + 6p +1 Which has the nice symmetrical look of a perfect square. The first and last terms make it fairly easy to spot the factorisation: p??? + 6p?? +11p?? + 6p +1 = (p?? + 3p + 1)?? Then I had a look at the roots: p(p + 3) + 1 = p?? + 3p + 1 (neat) but (p + 2)(p + 3) ??? 1 = p?? + 5p + 5, so I think there???s a typo in there, the other root should be (p + 1)(p + 2) ??? 1 (which is still neat)

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