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17 digits is too big.
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Gerhard Paseman
Gerhard Paseman
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Here is some (unverified) computational data, which I encourage others to extend.

I look at the squarefree p-smooth numbers (being $2^k$ in number for $p$ being the $k$th prime) and compared successive ratios. To reduce print out and operation cycles, I replaced the comparison $$\frac{x+a}{x} \gt \frac{y+b}{y}$$ with $$ay \gt xb$$, and whenever I got a better fraction I printed out in order $p : y+b, y, (y+b)/y, b, a$ . Here is the output up to p=71:

5 : 3 2 1.5 1 1
5 : 6 5 1.2 1 1
7 : 7 6 1.16667 1 1
7 : 15 14 1.07143 1 1
11 : 22 21 1.04762 1 1
13 : 66 65 1.01538 1 1
13 : 78 77 1.01299 1 1
17 : 715 714 1.0014 1 1
29 : 2002 2001 1.0005 1 1
29 : 2262 2261 1.00044 1 1
31 : 7163 7161 1.00028 2 1
31 : 12122 12121 1.00008 1 2
41 : 50065 50061 1.00008 4 1
41 : 82621 82615 1.00007 6 4
41 : 101270 101269 1.00001 1 6
43 : 958341 958334 1.00001 7 1
47 : 310247 310245 1.00001 2 7
47 : 487578 487577 1 1 2
47 : 2162095 2162094 1 1 1
53 : 21894574 21894565 1 9 1
61 : 66965190 66965177 1 13 9
61 : 1669770410 1669770141 1 269 13
67 : 118885413 118885403 1 10 269
67 : 4432525097 4432524801 1 296 10
71 : 62296466 62296465 1 1 296
71 : 6768250181 6768250086 1 95 1
71 : 16487003968153872 16487003736740238 1 231413634 95

I find the last line a rather startling indicator as to the problem difficulty. I suspect that it will be impossible to remove $\log$ from the exponent, and harder to prove that you cannot remove it.

EDIT: Unfortunately, the last values for $y$ have more decimal digits than the square root of the primorial for 71, leading me to suspect machine error. Even without that line though, the fact that there are many near-optimal ratios with difference larger than 1 presents some challenge. END EDIT.

Gerhard "Let's Do It Right Away" Paseman, 2015.04.28

Here is some (unverified) computational data, which I encourage others to extend.

I look at the squarefree p-smooth numbers (being $2^k$ in number for $p$ being the $k$th prime) and compared successive ratios. To reduce print out and operation cycles, I replaced the comparison $$\frac{x+a}{x} \gt \frac{y+b}{y}$$ with $$ay \gt xb$$, and whenever I got a better fraction I printed out in order $p : y+b, y, (y+b)/y, b, a$ . Here is the output up to p=71:

5 : 3 2 1.5 1 1
5 : 6 5 1.2 1 1
7 : 7 6 1.16667 1 1
7 : 15 14 1.07143 1 1
11 : 22 21 1.04762 1 1
13 : 66 65 1.01538 1 1
13 : 78 77 1.01299 1 1
17 : 715 714 1.0014 1 1
29 : 2002 2001 1.0005 1 1
29 : 2262 2261 1.00044 1 1
31 : 7163 7161 1.00028 2 1
31 : 12122 12121 1.00008 1 2
41 : 50065 50061 1.00008 4 1
41 : 82621 82615 1.00007 6 4
41 : 101270 101269 1.00001 1 6
43 : 958341 958334 1.00001 7 1
47 : 310247 310245 1.00001 2 7
47 : 487578 487577 1 1 2
47 : 2162095 2162094 1 1 1
53 : 21894574 21894565 1 9 1
61 : 66965190 66965177 1 13 9
61 : 1669770410 1669770141 1 269 13
67 : 118885413 118885403 1 10 269
67 : 4432525097 4432524801 1 296 10
71 : 62296466 62296465 1 1 296
71 : 6768250181 6768250086 1 95 1
71 : 16487003968153872 16487003736740238 1 231413634 95

I find the last line a rather startling indicator as to the problem difficulty. I suspect that it will be impossible to remove $\log$ from the exponent, and harder to prove that you cannot remove it.

Gerhard "Let's Do It Right Away" Paseman, 2015.04.28

Here is some (unverified) computational data, which I encourage others to extend.

I look at the squarefree p-smooth numbers (being $2^k$ in number for $p$ being the $k$th prime) and compared successive ratios. To reduce print out and operation cycles, I replaced the comparison $$\frac{x+a}{x} \gt \frac{y+b}{y}$$ with $$ay \gt xb$$, and whenever I got a better fraction I printed out in order $p : y+b, y, (y+b)/y, b, a$ . Here is the output up to p=71:

5 : 3 2 1.5 1 1
5 : 6 5 1.2 1 1
7 : 7 6 1.16667 1 1
7 : 15 14 1.07143 1 1
11 : 22 21 1.04762 1 1
13 : 66 65 1.01538 1 1
13 : 78 77 1.01299 1 1
17 : 715 714 1.0014 1 1
29 : 2002 2001 1.0005 1 1
29 : 2262 2261 1.00044 1 1
31 : 7163 7161 1.00028 2 1
31 : 12122 12121 1.00008 1 2
41 : 50065 50061 1.00008 4 1
41 : 82621 82615 1.00007 6 4
41 : 101270 101269 1.00001 1 6
43 : 958341 958334 1.00001 7 1
47 : 310247 310245 1.00001 2 7
47 : 487578 487577 1 1 2
47 : 2162095 2162094 1 1 1
53 : 21894574 21894565 1 9 1
61 : 66965190 66965177 1 13 9
61 : 1669770410 1669770141 1 269 13
67 : 118885413 118885403 1 10 269
67 : 4432525097 4432524801 1 296 10
71 : 62296466 62296465 1 1 296
71 : 6768250181 6768250086 1 95 1
71 : 16487003968153872 16487003736740238 1 231413634 95

I find the last line a rather startling indicator as to the problem difficulty. I suspect that it will be impossible to remove $\log$ from the exponent, and harder to prove that you cannot remove it.

EDIT: Unfortunately, the last values for $y$ have more decimal digits than the square root of the primorial for 71, leading me to suspect machine error. Even without that line though, the fact that there are many near-optimal ratios with difference larger than 1 presents some challenge. END EDIT.

Gerhard "Let's Do It Right Away" Paseman, 2015.04.28

Source Link
Gerhard Paseman
Gerhard Paseman
  • 3.2k
  • 1
  • 18
  • 29

Here is some (unverified) computational data, which I encourage others to extend.

I look at the squarefree p-smooth numbers (being $2^k$ in number for $p$ being the $k$th prime) and compared successive ratios. To reduce print out and operation cycles, I replaced the comparison $$\frac{x+a}{x} \gt \frac{y+b}{y}$$ with $$ay \gt xb$$, and whenever I got a better fraction I printed out in order $p : y+b, y, (y+b)/y, b, a$ . Here is the output up to p=71:

5 : 3 2 1.5 1 1
5 : 6 5 1.2 1 1
7 : 7 6 1.16667 1 1
7 : 15 14 1.07143 1 1
11 : 22 21 1.04762 1 1
13 : 66 65 1.01538 1 1
13 : 78 77 1.01299 1 1
17 : 715 714 1.0014 1 1
29 : 2002 2001 1.0005 1 1
29 : 2262 2261 1.00044 1 1
31 : 7163 7161 1.00028 2 1
31 : 12122 12121 1.00008 1 2
41 : 50065 50061 1.00008 4 1
41 : 82621 82615 1.00007 6 4
41 : 101270 101269 1.00001 1 6
43 : 958341 958334 1.00001 7 1
47 : 310247 310245 1.00001 2 7
47 : 487578 487577 1 1 2
47 : 2162095 2162094 1 1 1
53 : 21894574 21894565 1 9 1
61 : 66965190 66965177 1 13 9
61 : 1669770410 1669770141 1 269 13
67 : 118885413 118885403 1 10 269
67 : 4432525097 4432524801 1 296 10
71 : 62296466 62296465 1 1 296
71 : 6768250181 6768250086 1 95 1
71 : 16487003968153872 16487003736740238 1 231413634 95

I find the last line a rather startling indicator as to the problem difficulty. I suspect that it will be impossible to remove $\log$ from the exponent, and harder to prove that you cannot remove it.

Gerhard "Let's Do It Right Away" Paseman, 2015.04.28