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Main Question or Discussion Point
I came across something from the higher classes of math and I tried to prove it:
What do you call this kind of reasoning. Can it be called a proof? Or is this kind of like a pseudoproof? I've never seen any proofs actually. Is my reasoning correct or is it too ugly to look at?
[The illustrations are in the figure.png file.]
Prove that adding 1 to the product of four following natural numbers results in a square number.
Example: 1*2*3*4+1 = 25 or 5*6*7*8+1 = 1681
[tex]b^2 = a_1*a_2*a_3*a_4+1[/tex]
[tex]b = \sqrt{a_1*a_2*a_3*a_4+1}[/tex]
[tex]b^21 = a_1*a_2*a_3*a_4[/tex]
If ([tex]a_1*a_2*a_3*a_4[/tex]) was to be a square number without adding the one (+1), we could illustrate the product of multiplication with a geometrical square [Figure
#1]. But since we need to add (+1), it means that the square is incomplete (look at [Figure #2]).
The incomplete square in [Figure #2] is calculated to b^21 (look at the previous definitions of (b)). How would we calculate the area like in [Figure #1]? We factorize/break out (sorry, English is not my native tongue and I don't know the English math terminology):
[tex]\frac{b^21}{b1}[/tex] = (b+1) turns into [tex](b+1)(b1) = b^21[/tex]
Now we see that it is incomplete (if you look at Fig. #2).
What is taken to make [Figure #2] incomplete? We look at the difference:
[tex](b^2)(b^21) = 1[/tex]
If we neutralize it with (+1), we get a square number:
[tex]b^21+1 = b^2[/tex]
Besides, we know that adding +1 makes b^2 a square number (look at the third definition of b):
[tex]b^21+1 = b^2[/tex]
ABV
What do you call this kind of reasoning. Can it be called a proof? Or is this kind of like a pseudoproof? I've never seen any proofs actually. Is my reasoning correct or is it too ugly to look at?
[The illustrations are in the figure.png file.]
Prove that adding 1 to the product of four following natural numbers results in a square number.
Example: 1*2*3*4+1 = 25 or 5*6*7*8+1 = 1681
[tex]b^2 = a_1*a_2*a_3*a_4+1[/tex]
[tex]b = \sqrt{a_1*a_2*a_3*a_4+1}[/tex]
[tex]b^21 = a_1*a_2*a_3*a_4[/tex]
If ([tex]a_1*a_2*a_3*a_4[/tex]) was to be a square number without adding the one (+1), we could illustrate the product of multiplication with a geometrical square [Figure
#1]. But since we need to add (+1), it means that the square is incomplete (look at [Figure #2]).
The incomplete square in [Figure #2] is calculated to b^21 (look at the previous definitions of (b)). How would we calculate the area like in [Figure #1]? We factorize/break out (sorry, English is not my native tongue and I don't know the English math terminology):
[tex]\frac{b^21}{b1}[/tex] = (b+1) turns into [tex](b+1)(b1) = b^21[/tex]
Now we see that it is incomplete (if you look at Fig. #2).
What is taken to make [Figure #2] incomplete? We look at the difference:
[tex](b^2)(b^21) = 1[/tex]
If we neutralize it with (+1), we get a square number:
[tex]b^21+1 = b^2[/tex]
Besides, we know that adding +1 makes b^2 a square number (look at the third definition of b):
[tex]b^21+1 = b^2[/tex]
ABV
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