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anonymous

  • one year ago

For how many positive integers \(n\)\[\left \lfloor \frac{n^2}{3} \right \rfloor\]is a prime number? \(\left \lfloor \right \rfloor\) denotes the floor function.

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  1. ganeshie8
    • one year ago
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    my initial guess is to use little fermat for \((n,3)=1\) we have \[n^2\equiv 1\pmod{3}\]

  2. anonymous
    • one year ago
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    what comes after that?

  3. ganeshie8
    • one year ago
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    \[\left \lfloor \frac{n^2}{3} \right \rfloor = \left \lfloor \frac{3k+1}{3} \right \rfloor = k\] ?

  4. anonymous
    • one year ago
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    well, that's right, but how do we know that \(k\) is a prime or not?

  5. anonymous
    • one year ago
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    Use elementary methods. Just simplify the expression.

  6. ganeshie8
    • one year ago
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    \[n^2 = 3k+1 \implies 3k = n^2-1=(n-1)(n+1)\]

  7. P0sitr0n
    • one year ago
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    For a general field \[\mathbb{F}=\mathbb{R}\] this simplification won't work tho , since the floor function has an input interval that equals all the same value, i.e. \[\lfloor5.4\rfloor=\lfloor5.2\rfloor \nrightarrow 5.4 = 5.2\], so careful

  8. ganeshie8
    • one year ago
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    that yields \(\large k\in \{3,5\}\) consequently \(n\in \{4\}\)

  9. ganeshie8
    • one year ago
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    for \((n,3)\ne 1\), \(\left \lfloor \frac{n^2}{3} \right \rfloor =\left \lfloor \frac{(3k)^2}{3} \right \rfloor = 3k^2\) is always composite except for \(k=1\)

  10. ganeshie8
    • one year ago
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    Overall \(n\in\{3,4\}\)

  11. anonymous
    • one year ago
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    how about \(n=3k+2\)

  12. anonymous
    • one year ago
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    and \(n=3k+1\)?

  13. anonymous
    • one year ago
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    sry for late response, I was out

  14. ganeshie8
    • one year ago
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    little fermat covers both n=3k+1, 3k+2 right

  15. anonymous
    • one year ago
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    right :)

  16. ganeshie8
    • one year ago
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    would love to see an alternate method as my method above is pretty hacky

  17. ganeshie8
    • one year ago
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    another way to look at it is : \(n^2\) can never be \(3k+2 \) because \((3k\pm 1)^2 = 3M+1\). so \(n^2\equiv 0,1\pmod{3}\)

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