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 h
istorical 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

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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".

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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

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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)"

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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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