Problems & Puzzles: Puzzles

Puzzle 29.- Pi = P i-1&nxtprm(P i-1), Pi = prime for i => 1

Patrick De Geest sent this puzzle at September 10, 1998. In its own words:

"Step 1.
=======
Concatenation of prime p and nextprime(p) is a prime. (See A030459)

E.g. I.
smallest prime p = 2
nextprime(p) = 3
concatenate 2 and 3 and we get prime 23 [length : 2]

E.g. II. prime p = 5297
nextprime(p) = 5303
prime concatenation = 52975303 [length : 8]

Not so difficult I hear you utter and indeed larger examples are easy to find. So we continue with the next step. After all, we are warmed up now.

Step 2.
=======
Take a prime concatenation from step 1 and repeat the procedure exactly as described previously.

E.g. I. (step 1)
smallest prime p1 = 467
nextprime(p1) = 479
prime concatenation = 467479 --> becomes prime p2 !
(step 2)
prime p2 = 467479
nextprime(p2) = 467491
prime concatenation = 467479467491 [length : 12]

E.g. II. (step 1)
prime p1 = 25457
nextprime(p1) = 25463
prime concatenation = 2545725463 = prime p2
(step 2)
prime p2 = 2545725463
nextprime(p2) = 2545725499
prime concatenation = 25457254632545725499 [length : 20]

467 is the first one. The full series goes like :
467, 941, 13681, 14461, 21787, etc.
The prime concatenations quickly reach considerable lengths. But finding and testing them is still possible. That's why I propose to move on to step 3. As a coincidence our prime 467 is also the smallest prime that allows three steps before reaching a prime concatenation.

Step 3.
=======
E.g. I. see {Step 2 -> E.g. I} for the first two steps.
(step 3)
prime p3 = 467479467491
nextprime(p3) = 467479467527
prime concatenation = 467479467491467479467527 [length : 24]

E.g. II. (step 1)
prime p1 = 959941
nextprime(p1) = 959947
prime concatenation = 959941959947 = prime p2
(step 2)
prime p2 = 959941959947
nextprime(p2) = 959941959953
prime concatenation = 959941959947959941959953
(step 3)
prime p3 = 959941959947959941959953
nextprime(p3) = 959941959947959941959989
prime concatenation =
959941959947959941959953959941959947959941959989
[length : 40]

The first three primes that make a cycle of three steps possible are :
467, 941 and 959941
Now the level of difficulty reaches major proportions. Is there a prime that gives us a cycle of 4 steps ? The answer is yes.

Step 4.
=======
The smallest prime number that gives a cycle of four steps is '1936227'. The final prime concatenation has length 112 !
1936227 is up to now the only prime I found and tested OK ! The composite '7335450' is the second one in this series but then again, it is a composite number that we start the steps with, so not suitable for the puzzle.

(step 1)
smallest prime p1 = 1936227
nextprime(p1) = 1936237
prime concatenation = 19362271936237 = prime p2
(step 2)
nextprime(p2) = 19362271936243
prime concatenation = 1936227193623719362271936243 = prime p3
(step 3)
nextprime(p3) =1936227193623719362271936273
prime concatenation =
19362271936237193622719362431936227193623719362271936273
= prime p4
(step 4)
nextprime(p4 =
19362271936237193622719362431936227193623719362271936447
prime concatenation =
19362271936237193622719362431936227193623719362271936273/
19362271936237193622719362431936227193623719362271936447
[length : 112]

After this adventure till step 4, I need help from my puzzle comrades, to boldly go where no...
Is there a prime that gives us a cycle of 5 steps ? I don't know yet.
So, it's time to pose some final puzzle questions :

| P U Z Z L E Q U E S T I O N S .

| A.
|
| Find six more prime numbers that give cycles of three steps.
|
| 1. 467
|
| 2. 941
|
| 3. 959941
|
| 4. ?
|
| ...
|
| 9. ?
|
|
|
| B.
|
| Find two following prime numbers that give cycles of four steps.
|
| 1. 1936227
|
| 2. ?
|
| 3. ?
|
|
|
| C.
|
| Find the smallest prime numbers that gives cycles of '5 and more' steps".

Question A " Find six more prime numbers that give cycles of three steps"

Jo Yeong Uk has sent us an email (16/11/98) that says: "I've tested prime numbers that give cycles of any steps up to 1.8*10^7,and I found 15 numbers that give cycles of 3 steps bigger than 959941. The number is 3396199, 4858943, 5696101 ,6475643, 7566133, 7584253, 7592261 ,9305281, 9463877, 11430491, 13442243, 14374837, 15941473, 17414497, 17691997"

Question B "Find a prime number that gives a cycle of four steps"

Again Jo Yeong Uk discovered at 17/11/98 that the first prime number that gives a cycle of four steps is 127787377 (the first reported by Patrick De Geest - 1936227 - was not a prime number...)... and today 22/11/98 he reports the second one: 1510818931.Finnally today, 29/11/98 he sent the third, fouth and fifth ones: 3147482977, 3307903909 and 3621408103

***

I have made my own program to find and/or verify the results for this puzzle and this is the concatenated prime resulting from the 127707377 found by Jo:

127787377 (initial number)

127787377127787431 (Step 1)

127787377127787431127787377127787497 (step 2)

127787377127787431127787377127787497127787377127787431127787377127787503 (step 3)

127787377127787431127787377127787497127787377127787431127787377127787503

127787377127787431127787377127787497127787377127787431127787377127787641

(step 4)

***

On May 13, 2020, Giovanni Resta wrote:

200603842261 is smallest start of 5 steps: (I
don't think that "cycle" in this context is the right word).

Up to 1.2*10^13 there are 14 other such
numbers ( 788738880533, 1321335968963, 1372351933987,...) but not one
that generates a further step.

The primes generated by 200603842261 are:

200603842261200603842273

200603842261200603842273200603842261200603842333

200603842261200603842273200603842261200603842333
200603842261200603842273200603842261200603842537

p=
200603842261200603842273200603842261200603842333
200603842261200603842273200603842261200603842537
200603842261200603842273200603842261200603842333
200603842261200603842273200603842261200603842843

And the next and last it the concatenation of p above with p+504 (the
next prime).

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