I like spotting patterns and came across the fact that 3 is the only prime that comes before a square number (3, 4)
I can prove this by thinking about factorising x^2 -1 .
I also spotted that 2 is the only prime number which is 1 more than a cube number ( 1, 2). And the proof involves looking at factorising x^3+1. Again i can do this.
I recently saw this claim ( on someones twitter!)
25 is the only square number that can be expressed as cube minus 2 ?
25 = 3^3-2
Now, i am stuck for ideas on this and it feels like a different beast compared to the first two examples.
Any tips? My only thought was exploring how i could show that y^2 = x^3 -2 only has a solution for a=3, b= 5 ??? If indeed that is the case?
I can prove this by thinking about factorising x^2 -1 .
I also spotted that 2 is the only prime number which is 1 more than a cube number ( 1, 2). And the proof involves looking at factorising x^3+1. Again i can do this.
I recently saw this claim ( on someones twitter!)
25 is the only square number that can be expressed as cube minus 2 ?
25 = 3^3-2
Now, i am stuck for ideas on this and it feels like a different beast compared to the first two examples.
Any tips? My only thought was exploring how i could show that y^2 = x^3 -2 only has a solution for a=3, b= 5 ??? If indeed that is the case?
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