Problems & Puzzles:
Conjectures
Conjecture 18.- Minimal
primorial partitions (a conjecture by John Harvester)
Let's define p(n) as the product of the first n primes
(this is the so-called "primorial function" sometimes symbolized
by p#)
Now let's suppose that for each n, we find those X
& Y such that:
- X*Y = p(n)
- (X-Y) = minimal, X>Y
Here are some values for n = 1 to 15
| n |
p(n) |
X |
Y |
min(X-Y) |
| 1 |
2 |
2 |
1 |
2-1=1 |
| 2 |
6 |
3 |
2 |
3-2=1 |
| 3 |
30 |
6 |
5 |
=2*3-5=1 |
| 4 |
210 |
15 |
14 |
=3*5-2*7=1 |
| 5 |
2310 |
55 |
42 |
5*11-2*3*7=13 |
| 6 |
|
|
|
17 |
| 7 |
|
|
|
1 |
| 8 |
|
|
|
41 |
| 9 |
|
|
|
157 |
| 10 |
|
|
|
1811 |
| 11 |
|
|
|
1579 |
| 12 |
|
|
|
18859 |
| 13 |
|
|
|
95533 |
| 14 |
|
|
|
17659 |
| 15 |
|
|
|
1995293 |
John Harvester conjectures that:
min(X-Y)<e^n.
1. Can you fill the blank cells of
the table up to n=15?.
2. Can you propose an efficient algorithm to find the X & Y values?
3. Can you verify the conjecture for
100>n>15?
4. Can you propose a "better" bound?
__________
Some historical notes:
Erdös conjectured that min (X-Y)=1 for only
n=1, 2, 3, 4 & 7. This has been verified recently by Chris
Nash up to n=600000. As far as John knows
his conjecture has never been posted before.
Solution
Jud McCranie has calculated (15/03/2000) X &
Y for n<=25 and says: "conjecture 18 fails for n=19, 20, 21,
23, 24, 25. For instance, for n=19, min x-y = 299,808,329 > e^19 =
178,482,300."
| n |
X |
Y |
min (X-Y) |
| 19 |
2803419704514 |
2803119896185 |
299808329 |
| 20 |
23622001517543 |
23619540863730 |
2460653813 |
| 21 |
201817933409378 |
201813981102615 |
3952306763 |
| 23 |
16342166369958702 |
16342050964565645 |
115405393057 |
| 24 |
154171363634898185 |
154170926013430326 |
437621467859 |
| 25 |
1518410187442699518 |
1518409177581024365 |
1009861675153 |
***
Jud conjectures that min
(X-Y)<c*sqrt[p(n)/e^n], where "c
on the order of 1...". (As
a matter of fact for n<=33, c<5 works fine) "...c may
-> 1 as n-> infinity, but I don't "believe" that yet. I
don't have any reason for believing it, except that it would be nice".
***
Other values obtained by Jud are:
26
x = 3*5*7*11*13*19*23*31*41*47*61*71*89*101
y = 2*17*29*37*43*53*59*67*73*79*83*97
min is 6660853109087
27
x = 2*3*5*7*11*17*23*41*43*47*59*67*71*73*101
y = 13*19*29*31*37*53*61*79*83*89*97*103
min is 29075165225531
28
x = 3*5*13*17*19*29*41*53*61*79*83*97*101*103
y = 2*7*11*23*31*37*43*47*59*67*71*73*89*107
min is 418895584426457
29
x = 2*3*7*19*23*29*37*41*43*61*83*89*97*103*107
y = 5*11*13*17*31*47*53*59*67*71*73*79*101*109
min is 2371362636817019
30
x = 2*7*11*23*29*37*43*47*59*61*79*83*89*103*107
y = 3*5*13*17*19*31*41*53*67*71*73*97*101*109*113
min is 6889206780487667
31
x = 2*3*5*7*11*17*29*31*37*47*61*67*73*83*107*109*113
y = 13*19*23*41*43*53*59*71*79*89*97*101*103*127
min is 5258351738694673
32
x = 3*13*17*19*47*61*71*79*83*89*97*101*109*113*127
y = 2*5*7*11*23*29*31*37*41*43*53*59*67*73*103*107*131
min is 19193968546850561
33
x = 3*5*7*17*23*29*37*41*43*47*59*71*101*103*109*113*137
y = 2*11*13*19*31*53*61*67*73*79*83*89*97*107*127*131
X-Ymin 1157424575582244547
34
x = 2*7*13*19*31*37*41*43*67*71*79*83*89*97*107*113*139
y = 3*5*11*17*23*29*47*53*59*61*73*101*103*109*127*131*137
X-Ymin 18534732682906712369
35
x = 3*7*29*31*37*43*47*59*61*67*73*79*89*97*127*131*137
y = 2*5*11*13*17*19*23*41*53*71*83*101*103*107*109*113*139*149
X-Ymin 44413798987295791
36
x = 2*5*7*13*29*37*41*61*67*71*89*97*107*113*127*137*149*151
y = 3*11*17*19*23*31*43*47*53*59*73*79*83*101*103*109*131*139
X-Ymin 427009256730780749983
37
x = 2*5*7*31*41*43*61*71*73*89*103*107*109*113*127*137*149*157
y = 3*11*13*17*19*23*29*37*47*53*59*67*79*83*97*101*131*139*151
X-Ymin 1288425690589837107401
***
Regarding other proposals of bounds, Jud sent the
following ones:
"1. For n, (x-y)min <= c*e^((Pn-n)/2) where Pn is the nth
prime, c is a constant.
2. If that doesn't hold, then (x-y)min <= c*e^(Pn/2)"
***
John Harvester wrote (25 & 26/04/2000):
"Mr McCranie's latest results for min(x-y)
suggest, in my view strongly, the relationship log(min(x-y)) ~ (p(n))^(1/n),
where p(n) is the primorial function. Alternatively, given that (p(n))^(1/n)
-> (Pn/e) as n -> oo, where Pn denotes the n-th prime, it might turn
out that log(min(x-y)) ~ Pn/e....The limit is based on the relation (theta(x)/pi(x))-log(x)
~ -1, where x is the n-th prime. I understand that the proof is given in Landau's
Primzahlen and would refer you to the item from Stephen Finch's
"Favourite Mathematical Constants" Pages, at: http://www.mathsoft.com/asolve/constant/e/eprime.html.
I'd be pleased if this limit conjecture for min(x-y) were true.
Given that sqrt(p(n)) ~ e^(Pn/2), Jud's conjecture that min(x-y) is
bounded by e^(Pn/2) is probably right but it should be possible to find a
stronger one. Having thought about it overnight, e^(Pn/e) seems like a
plausible candidate"
So, the new Harvester's conjecture is: min(x-y)<e^(Pn/e).
Is this true? Is this the "best" upper bound?
***
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