Problems & Puzzles: Puzzles

Puzzle 141.  Smarandache Prime Base representation of the natural numbers

María T. Marcos from Philippines, some days ago sent an unsolved and very nice problem related to the Smarandache Prime Base (SPB) representation of the natural  numbers.

The SBP is the ordered infinite set of the prime numbers proceeded by the number 1:

SBP={1, 2, 3, 5, 7, 11, 13, ...}

A formal and rigorous explanation of a "representation" of any natural number n in the SPB can be found here, but I will explain this "representation" with an example:

Let's suppose that n = 27, then you partition 27 as a sum of the prime less or equal to 27 plus its complement (n = p + c):

27 = 23 + 4

If the complement is "0" or any of the members of the SPB set you have done it; if not you repeat and find the partition to the complement:

4 =3 +1 (Done!)

Then 27 = 23 +3 +1, and the representation of 27 in the SBP is a string of bits informing you what primes enter in the partition of n:

27 = 1x23+0x19+0x17+0x11+0x7+0x5+1x3+0x2+1x1 = 100000101

Evidently the k-th prime number is represented in this system as 1(0)k.

The representation of the first 27 natural numbers in the SPB system is this:

0, 1, 10, 100, 101, 1000, 1001, 10000, 10001, 10010, 10100, 100000, 100001, 1000000, 1000001, 1000010, 1000100, 10000000, 10000001, 100000000, 100000001, 100000010, 100000100, 1000000000, 1000000001, 1000000010, 1000000100, 1000000101 (primes are in bold type)

The María's question is this:

1. What is the integer with the largest number of digits "1" in this base?


By my own, I have calculated the earliest numbers n that uses K digits "1":

K, quantity of digits "1" in the SPB representation of n Earliest n(K) Partition
0 0 0=0
1 1 1=1
2 4 4 =3+1
3 27 27=23+3+1
4 1354 1354 =1327+23+3+1
5 ?  

2. Can you extend the table?

In the meanwhile that someone can get the earliest n(K) I would like to advance two things:

A.- Theoretically n(5) should be around 1.7962*10^17.
3a) Can you explain where comes from this value, or provide another estimate?

B.- I have gotten one solution for K=5, but definitively NOT the earliest because the n value obtained has 102 digits against the 18 digits forecasted. The only purpose of this solution is to show that solutions exist, and to show a way of effectively calculating n(5): 

  • n = p+1327+23+3+1, such that
  • p = prime, digits(p) = 102
  • n - p = 1354
  • There is not any prime between n & p (as a matter of fact the total prime gap after p is 1608)

3b) Can you devise how I calculated p and/or improve it (to get a smaller solution for K=5)?

4. Suppose that someone obtains the real earliest n(5) and that its value is according to the theoretical forecast (n(5) ~ 1.7962*10^17 ). Can you estimate the magnitude order of n(6)? 

N. B: Last but not least: Gentlemen, please say all of us Welcome! to María T. Marcos, the first woman puzzler of our pages.



[1] Dumitrescu, C., Seleacu, V., "Some notions and questions in number theory", Xiquan Publ. Hse., Glendale, 1994, Sections #47-51; 

[2] Grebenikova, Irina, "Some Bases of Numerations", <Abstracts of Papers Presented at the American Mathematical Society>, Vol. 17, No. 3, Issue 105, 1996, p. 588.

[3] Smarandache Bases, 



Regarding the Maria's question (1) Chris Nash wrote:

" is certain n(k) exists for every k, we just need to find a big enough gap (and of course we can use things like n!+2, n!+3..... n!+n to know gaps of any size do exist and so a gap exists that is bigger than the previousn(k)).

A similar idea was expressed by Key Toshihara from Osaka, Japan:

"...there are numbers with as many as we want digits of "1". I base my reasoning on the fact that fact that the gap between any two consecutive prime can be as much as we want..."

Chu Lai, from Beijing, China wrote simply: "The problem has infinitely many solutions."


Regarding the theoretical estimate for n(5)- question 3a - Chris Nash and Jud McCranie argued basically the same:

"The number for k=5 must be after a prime gap > 1354. (If it were a smaller gap, the number minus the largest prime less than it would be < 1354, so less than four more primes would be needed to represent it, so it wouldn't require five primes.) There is an estimate of where the first gap of size G occurs - e^(r*sqrt(G)), where r is a constant about 1. For gaps currently known, r is about 1.1 to 1.13. Using G=1356, r=1.1 to 1.13, we get an estimate of 3.9E17 to 1.2E18." (Jud)

A similar reasoning come from Felice Russo.

But only Chris Nash discovered how I calculated my solution for K=5 (Question 3.b):

"...When I wrote last time I did not see where it came from. Then I saw all the 9's and I thought to round up to the number ending in many zeroes.... which is N=2^67*3^32*5^16*7^11*11^6*13^5*17^4*19^3*23^3*29^2*31^2*37*41*43*47*53 *59*61*67*71 Oh! I think I see it :) That is 71! So we test N-1 and N+1 (they are composite) and so we certainly are in a large prime gap (*at* least from N-71 to N+71). So then we seek for two primes N-a and N+b to work out the exact gap size? I am guessing you tested all the factorials up to 71 and finally this one produced a big enough gap... That gives me some search ideas :)"



The order of magnitude of n(6) - question 4 - was estimated by Jud and Chris:

"...Well, I'll take n(5)=10^16 as an easy estimate and now what's needed is the first prime gap of size 10^16, and so p ~ exp(sqrt(M)) = exp(10^8) which has over 43 million digits..." (Chris)

"...This gets really large, and the estimate is very imprecise because it is based on another estimate. But assuming that n(5) is about 3.9E17, we need a gap that large to have n(6), so it would probably be at least e^(sqrt(3.9E17)), a number with more than 271,000,000 digits." (Jud)

Felice Russo wrote "...the magnitude of term n(6) should be very big being: n(6)~exp(sqrt(1.165746*1.82222*10^17))


I asked to Chris for an estimate when - in the future - n(6) could be effectively calculated. This is his answer:

"I do not know. Unless there is new theory, finding just *one* prime of D digits takes time that grows faster than D^3. So it would take a million times as long. Perhaps in 30 years time our computers will be a million times faster, so perhaps in 2031 someone will find the *first known* two hundred million digit prime. But that is just for *one* prime!

But finding these gaps (and finding the smallest prime with a given gap) is *much* harder (because all the other gaps before it need to be checked?). Today I think the gap tables are only complete to about 1100, and this seem to takes time that grows with N, not with digits - so who knows, perhaps even the *smallest* n(5) solution is difficult. We may have to settle for *the smallest known*"


So, after all these smart collaborations the only question remaining is to calculate the earlier n(5). For this we need really brave people. Of course that we may accept - in the mean while - smaller solutions than the one I got.


But the SPB has some attraction here and there.

Key Toshihara asked "...find numbers in Smarandache prime base whose digits are equal to "1" only, what is the largest such number...".

Motua Guang, form Cairo, Egypt ask for "...the biggest pierced number in the SBP..." [a pierced number is any number 1(01)n, n=odd ].

I believe that after the discussion above over the numbers n(k), easy answers can be given to both questions.


Charles Greathouse wrote (Aug, 2010):

1. The fifth term is 401429925999155061 = 401429925999153707 + 1327 + 23 + 3 + 1. This was independently calculated in 2007 by Richard Mathar and Florian Luca & Ravindranathan Thangadurai, both using Thomas Nicely's tables.

See and Florian Luca & Ravindranathan Thangadurai, "On an arithmetic function considered by Pillai", Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.

2. The problem does not originate with Smarandache!  It was considered
by Pillai 80 years ago in his paper "An arithmetical function
concerning primes" (Annamalai University Journal (1930), pp. 159–167.)
3. In the above paper, Pillai answers María's question: arbitrarily
many primes may be needed.
4. Pillai proved several interesting results about the sequence not on
the page.  For example, O(log n) primes are needed to represent n.
5. Florian Luca & Ravindranathan Thangadurai (2009) improve on
Pillai's results.  In particular, O(log log n) primes are needed to
represent n.
6. On Cramer's conjecture (or Maier's version), the number of primes
required is O(log* n), where log* is the iterated logarithm.








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