Problems & Puzzles: Puzzles

Puzzle 75.- Prime numbers and the number 2000

I couldn’t resist the temptation of calculating some prime numbers related to the number 2000 (the conspicuous year coming). Below you can found six claims. I guess that the first three claims are ending absolute results, while the other three for sure may be improved.

1. Smallest prime with D = 2000 digits:
N = 10^1999 + 7321
= 1(0)1995 73211995 7321, SOD = 14
(Actually a strong pseudoprime)

2. Largest prime with D = 2000 digits:
N = 10^2000-
9297= (9)1996 07031996 0703, SOD = 17964
(Actually a strong pseudoprime)

3. Smallest prime such that SOD = 2000:
N =
3*10^222 +(10^222-1) –10^125 = 3(9)96 8(9)125 , D = 223
(Absolute prime verified with APRT-CLE)

4. Smallest calculated prime such that D = SOD = 2000:
N=10^1999+9*10^(1643+222)+(10^221-1)*10+1 =
1(0)
133 9 (0)1643 (9)221 1133 9 (0)1643 (9)221 1
(Actually a strong pseudoprime)

5. Smallest calculated prime k*2^n + 1 such that SOD=2000
N = 75727*2^1458 +1,
D=444

(Absolute prime verified with Proth.exe)

6. Smallest calculated prime k*2^n - 1 such that SOD=2000
N = 139119*2^1457 - 1,
D=444

(Absolute prime verified with Proth.exe)

Questions:

1.Would you like to improve the claims 4, 5 & 6?
2.
Can you redo the exercise for 1, 2, 3 & 4 using palprimes?
3.
I'll be very glad to add here other curio suggestions related with primes & the number 2000.

Chris Nash found "the smallest prime with 2000 digits and digit sum 2000. Here is his email sent at 29/11/99, solving completely the 4th claim of this puzzle:

"the smallest prime with 2000 digits and digit sum of 2000. It is 1(0)17763(9)78(9)968(9)117.

It is probable prime, and as yet not proven, but could be provable I am certain with a little extra factoring work on N+1. It is certainly the smallest such prime. It was discovered as follows. First the number

X = 1(0)
17763(9)222 = 10^1999 + 4*10^222 - 1
was constructed. It has sum of digits 2002. Then a search was done for
primes of the form

X - 10^(223-n) - 10^(223-k)
using PrimeForm and searching from n=1 to 223, k=1 to 223. Note that all numbers of this form have 2000 digits and digit sum 2000, finally note the first prime found by this search will be smaller than any other. The result followed shortly afterwards - n=9, k=106, giving the closed form

N = 10^1999 + 4*10^222 - 10^214 - 10^117 - 1.
"

Latter he added: "Looking at it again, I thought it very interesting (at least for Americans!) that this 2000 digit prime with sum of digits 2000 had precisely 1776 zeroes in it - it is also a '4th of July' prime as well as a 'millenium' prime..."

Chris Nash has also found smaller primes of the form k*2^n+/-1 such that SOD=2000:

33401*2^1373+1
67893*2^1371-1

Both with 418 digits.

"I used a modified version of PrimeForm which first tested the sum of digits, then trial factoring, finally primality proof... I'm going to attempt n=1350, but I doubt I will find an answer, the probability here is very very close to zero... there may be a smaller solution than these, but it will take a lot of searching I think!"

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On Jan 9, 2019 Pierandrea Formusa wrote:

Find below my answer to your Puzzle 75, with regard to question 3 (3.I'll be very glad to add here other curio suggestions related with primes & the number 2000. ).

We can express 2000 like the sum of a prime and the square of a prime and the cube of a prime and the square of the square of a prime in a prime number of ways (13), with all primes different. And 13 appear only 2 times considering all solutions and 2 is the only even prime.

Below the 13 solutions:
 p1 p2 p3 p4 p1 p2^2 p3^3 p4^4 sum 1433 19 5 3 1433 361 125 81 2000 953 29 5 3 953 841 125 81 2000 113 41 5 3 113 1681 125 81 2000 563 5 11 3 563 25 1331 81 2000 419 13 11 3 419 169 1331 81 2000 227 19 11 3 227 361 1331 81 2000 59 23 11 3 59 529 1331 81 2000 911 11 7 5 911 121 343 625 2000 863 13 7 5 863 169 343 625 2000 743 17 7 5 743 289 343 625 2000 503 23 7 5 503 529 343 625 2000 191 29 7 5 191 841 343 625 2000 71 31 7 5 71 961 343 625 2000

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On Jan 16, 2019, Paul Cleary wrote:

No.5.   89393068 2^1272+1, sod =2000 and 391 digits.

No.6.   8567418*2^1272-1, sod = 2000 and 390 digits.

I found approx. 50 or so other solutions between 444 digits and 390, they do seem to have dried up though smaller than 390 digits.

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