Using Least common multiple to establish a lower bound for a ratio of primorials
$begingroup$
Let $x#$ be the primorial for $x$.
Let $text{lcm}(x)$ be the least common multiple of ${1, 2, 3, dots, x}$.
It occurs to me that for $x ge 4$, it is straight forward to find a lower bound of $dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}$.
I find it interesting that an upper bound can lead to a lower bound.
Is my reasoning correct?
Here's my argument:
(1) $text{lcm}left(dfrac{x^2+x}{2}right)text{lcm}(x)left(dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > text{lcm}(x^2+x) $
if a prime $p ge (x+1)$, then $p^2 > (x^2+x)$
if a prime $(x^2+x) ge p^a > frac{x^2+x}{2}$, then $p^{a+1} > (x^2+x)$ and $p^{a-1} le frac{x^2+x}{2}$
(2) $text{lcm}(x^2+x) ge {{x^2+x}choose{frac{x^2+x}{2}}}$
This follows from Legendre's Formula since:
Let $v_p(x)$ be the highest power of $p$ that divides $x$
$v_p({{x^2+x}choose{frac{x^2+x}{2}}}) = sumlimits_{i ge 1}leftlfloordfrac{x^2+x}{p^i}rightrfloor - 2leftlfloordfrac{x^2+x}{2p^i}rightrfloor$
It is well known that for each $i$, the difference is at most $1$ and that if $i > log_p(x^2+x)$, then the difference is $0$.
(3) ${{x^2+x}choose{frac{x^2+x}{2}}} > dfrac{2^{x^2+x}}{frac{x^2+x}{2}}$
For $x ge 4$, ${{2x}choose{x}} ge dfrac{4^x}{x}$ since ${8choose4} = 70 > dfrac{4^4}{4} = 64$ and ${{2x}choose{x}} = 2left(dfrac{2x-1}{x}right){2(x-1)choose{x-1}} > 2left(dfrac{2x-1}{x}right)left(dfrac{4^{x-1}}{x-1}right) > dfrac{4^x}{x}$
(4) Using Hanson's result that $text{lcm}(x) < 3^x$, gives:
$$left(frac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > left(frac{4}{3}right)^{(x^2+x)/2}left(frac{2}{(x^2+x)3^x}right)$$
proof-verification prime-numbers upper-lower-bounds least-common-multiple
$endgroup$
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$begingroup$
Let $x#$ be the primorial for $x$.
Let $text{lcm}(x)$ be the least common multiple of ${1, 2, 3, dots, x}$.
It occurs to me that for $x ge 4$, it is straight forward to find a lower bound of $dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}$.
I find it interesting that an upper bound can lead to a lower bound.
Is my reasoning correct?
Here's my argument:
(1) $text{lcm}left(dfrac{x^2+x}{2}right)text{lcm}(x)left(dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > text{lcm}(x^2+x) $
if a prime $p ge (x+1)$, then $p^2 > (x^2+x)$
if a prime $(x^2+x) ge p^a > frac{x^2+x}{2}$, then $p^{a+1} > (x^2+x)$ and $p^{a-1} le frac{x^2+x}{2}$
(2) $text{lcm}(x^2+x) ge {{x^2+x}choose{frac{x^2+x}{2}}}$
This follows from Legendre's Formula since:
Let $v_p(x)$ be the highest power of $p$ that divides $x$
$v_p({{x^2+x}choose{frac{x^2+x}{2}}}) = sumlimits_{i ge 1}leftlfloordfrac{x^2+x}{p^i}rightrfloor - 2leftlfloordfrac{x^2+x}{2p^i}rightrfloor$
It is well known that for each $i$, the difference is at most $1$ and that if $i > log_p(x^2+x)$, then the difference is $0$.
(3) ${{x^2+x}choose{frac{x^2+x}{2}}} > dfrac{2^{x^2+x}}{frac{x^2+x}{2}}$
For $x ge 4$, ${{2x}choose{x}} ge dfrac{4^x}{x}$ since ${8choose4} = 70 > dfrac{4^4}{4} = 64$ and ${{2x}choose{x}} = 2left(dfrac{2x-1}{x}right){2(x-1)choose{x-1}} > 2left(dfrac{2x-1}{x}right)left(dfrac{4^{x-1}}{x-1}right) > dfrac{4^x}{x}$
(4) Using Hanson's result that $text{lcm}(x) < 3^x$, gives:
$$left(frac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > left(frac{4}{3}right)^{(x^2+x)/2}left(frac{2}{(x^2+x)3^x}right)$$
proof-verification prime-numbers upper-lower-bounds least-common-multiple
$endgroup$
add a comment |
$begingroup$
Let $x#$ be the primorial for $x$.
Let $text{lcm}(x)$ be the least common multiple of ${1, 2, 3, dots, x}$.
It occurs to me that for $x ge 4$, it is straight forward to find a lower bound of $dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}$.
I find it interesting that an upper bound can lead to a lower bound.
Is my reasoning correct?
Here's my argument:
(1) $text{lcm}left(dfrac{x^2+x}{2}right)text{lcm}(x)left(dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > text{lcm}(x^2+x) $
if a prime $p ge (x+1)$, then $p^2 > (x^2+x)$
if a prime $(x^2+x) ge p^a > frac{x^2+x}{2}$, then $p^{a+1} > (x^2+x)$ and $p^{a-1} le frac{x^2+x}{2}$
(2) $text{lcm}(x^2+x) ge {{x^2+x}choose{frac{x^2+x}{2}}}$
This follows from Legendre's Formula since:
Let $v_p(x)$ be the highest power of $p$ that divides $x$
$v_p({{x^2+x}choose{frac{x^2+x}{2}}}) = sumlimits_{i ge 1}leftlfloordfrac{x^2+x}{p^i}rightrfloor - 2leftlfloordfrac{x^2+x}{2p^i}rightrfloor$
It is well known that for each $i$, the difference is at most $1$ and that if $i > log_p(x^2+x)$, then the difference is $0$.
(3) ${{x^2+x}choose{frac{x^2+x}{2}}} > dfrac{2^{x^2+x}}{frac{x^2+x}{2}}$
For $x ge 4$, ${{2x}choose{x}} ge dfrac{4^x}{x}$ since ${8choose4} = 70 > dfrac{4^4}{4} = 64$ and ${{2x}choose{x}} = 2left(dfrac{2x-1}{x}right){2(x-1)choose{x-1}} > 2left(dfrac{2x-1}{x}right)left(dfrac{4^{x-1}}{x-1}right) > dfrac{4^x}{x}$
(4) Using Hanson's result that $text{lcm}(x) < 3^x$, gives:
$$left(frac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > left(frac{4}{3}right)^{(x^2+x)/2}left(frac{2}{(x^2+x)3^x}right)$$
proof-verification prime-numbers upper-lower-bounds least-common-multiple
$endgroup$
Let $x#$ be the primorial for $x$.
Let $text{lcm}(x)$ be the least common multiple of ${1, 2, 3, dots, x}$.
It occurs to me that for $x ge 4$, it is straight forward to find a lower bound of $dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}$.
I find it interesting that an upper bound can lead to a lower bound.
Is my reasoning correct?
Here's my argument:
(1) $text{lcm}left(dfrac{x^2+x}{2}right)text{lcm}(x)left(dfrac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > text{lcm}(x^2+x) $
if a prime $p ge (x+1)$, then $p^2 > (x^2+x)$
if a prime $(x^2+x) ge p^a > frac{x^2+x}{2}$, then $p^{a+1} > (x^2+x)$ and $p^{a-1} le frac{x^2+x}{2}$
(2) $text{lcm}(x^2+x) ge {{x^2+x}choose{frac{x^2+x}{2}}}$
This follows from Legendre's Formula since:
Let $v_p(x)$ be the highest power of $p$ that divides $x$
$v_p({{x^2+x}choose{frac{x^2+x}{2}}}) = sumlimits_{i ge 1}leftlfloordfrac{x^2+x}{p^i}rightrfloor - 2leftlfloordfrac{x^2+x}{2p^i}rightrfloor$
It is well known that for each $i$, the difference is at most $1$ and that if $i > log_p(x^2+x)$, then the difference is $0$.
(3) ${{x^2+x}choose{frac{x^2+x}{2}}} > dfrac{2^{x^2+x}}{frac{x^2+x}{2}}$
For $x ge 4$, ${{2x}choose{x}} ge dfrac{4^x}{x}$ since ${8choose4} = 70 > dfrac{4^4}{4} = 64$ and ${{2x}choose{x}} = 2left(dfrac{2x-1}{x}right){2(x-1)choose{x-1}} > 2left(dfrac{2x-1}{x}right)left(dfrac{4^{x-1}}{x-1}right) > dfrac{4^x}{x}$
(4) Using Hanson's result that $text{lcm}(x) < 3^x$, gives:
$$left(frac{(x^2+x)#}{left(frac{x^2+x}{2}right)#}right) > left(frac{4}{3}right)^{(x^2+x)/2}left(frac{2}{(x^2+x)3^x}right)$$
proof-verification prime-numbers upper-lower-bounds least-common-multiple
proof-verification prime-numbers upper-lower-bounds least-common-multiple
asked Dec 12 '18 at 3:49
Larry FreemanLarry Freeman
3,17521239
3,17521239
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