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Writer's pictureIstvan Benedek

Extension of the problem a*b is unique for a,b in [n^2... (n+1)^2-1]

I have recently found this mathematical statement of Pál Erdős:

If we take two arbitrary numbers, a and b, from the range of [n^2 ... (n+1)^2-1], then every a*b will be unique. I found the proof and started to think about what if we could extend this set with other ranges where k != n [k^2 ... (k+1)^2-1] ...


I had a weak conjecture about there should be some k1 k2 k3.... kn for every n where none of the ks equals to n and union lets denote with U of the following ranges [n^2 ... (n+1)^2-1],  [k1^2 ... (k1+1)^2-1] ..... [kn^2 ... (kn+1)^2-1] keeps the original behaviour :

a, b, c, d elements of U where a*b = c *d only and only if a = c and b = d.


I did not make the proof yet; I wanted to get a quick answer to my conjecture, so I created a small Python code, which showed interesting things.


I found that

[4^2 ... 5^2-1] denote with M4 fits with

[39^2 ... 40^2 -1] denote with M39


But the interesting thing was, it fit with every range between R39-R1000 except the R40.


Updated on 24 January 2024


I use M after the Mersene


M4 = [4^2 ... 5^2-1]

M39 = [39^2 ... 40^2 -1]

M3454 = [3454^2 ... 3455^2 -1]

M3456 = [3456^2 ... 3457^2 -1]


If M = M4 U M39 U M3454, then a, b, c, d elements of M where a*b = c *d only and only if a = c and b = d.


If M = M4 U M39 U M3456, then a, b, c, d elements of M where a*b = c *d only and only if a = c and b = d.


 

I use F after Fermat


F4 = [4^2+1 ... 5^2]

F41 = [41^2+1 ... 42^2]

F3454 = [3454^2+1 ... 3455^2]

F3456 = [3456^2+1 ... 3457^2]


F = F4 U F41 U F3456, then a, b, c, d elements of M where a*b = c *d only and only if a = c and b = d.


F = F4 U F41 U F3454, then a, b, c, d elements of M where a*b = c *d only and only if a = c and b = d or a,b,c,d =1763, 11936988, 1764, 11930221


1764 = 42^2

1763 = 42^2-1

So , F = F4 U (F3454 - {42^2}) U F3456), then a, b, c, d elements of M where a*b = c *d only and only if a = c and b = d.



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