Problems & Puzzles:
Puzzles
Puzzle 163. P+SOD(P)
Enoch
Haga sent the following puzzle:
Let's
start with a prime P, then let's add to P the sum of the digits of P.
The new number may be, or not, another prime number. If not, then repeat
the procedure until you get a prime number.
Example:
Starting
with P=2, the sequence generated is:
2,
4, 8, 16, 23 (prime & bingo!),
Large of the sequence, L = 5
Questions:
a)
Can L take any value?
b) Find the earliest primes corresponding to L=100, 200, 300, .. 1000
The
Enoch's puzzle let me think in some other similar questions: what
about if we use the same procedure to generate the members of the same sequences but now we add the condition that all the members must be
prime numbers ending when a composite number follows?
Example:
P=277, L=4: 277, 293, 307, 317 (all are prime numbers)
Here
are the earlier sequences of L members of this type, that I have obtained:
L 
First
prime of the sequence 
1 
2 
3 
11 
4 
277 
5 
37783 
6 
516493 
8 
286330897 
c)
Can L take any value?
d) Find the earliest prime for L=9, 10, 11, 12 & 13
Last,
what if we ask additionally that all the primes in the sequence need to be
consecutive primes.
Example:
P=11, L=3: 11, 13, 17
For
this case, the largest sequence that I have found is: P=1427411, L=4
e)
Find the earliest prime for L=5, 6 &7
Solution:
Sudipta
Das found, for the question b) that "the earliest prime P for L = 100 is 954977".
This same value was also found independently by JeanChristophe Colin.
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Sudipta
Das found, for the question b) that "the earliest prime P for L =
200 is 1306002569"
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Felice Russo wrote (28/1/02): "...for the puzzle 163
question d) I didn't find a prime with L>8 up to 18038439735."
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Felice Russo made a statistical analysis for the question c)
Can L take any value?
and found an interesting result: "The
experimental data seem to support that L cannot take any value and that
most likely the maximum value should be L=14"
If you want to read the Russo's complete analysis
you may download it clicking here.
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On 16/9/6 C. Rivera wrote:
Smallest solution to question e) for K=5:
317130731, 317130757, 317130791, 317130823, 317130851
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Giovanni Resta wrote on March 2011:
for question (d) of Puzzle 163 I found that
the first sequence with L=9 is
56676324799, 56676324863, 56676324919, 56676324977, 56676325039,
56676325091, 56676325141, 56676325187, 56676325243.
Other sequences with L=9 start at
373169411809, 2121959132809, 10180781225809, 14328311692789,
17429111275789, 32594135422789, 34327062247789, 39262151325799.
But I did not find (yet!) a sequence with L=10.
For question (e) of Puzzle 163 I found that
the earliest sequences with L=6,7 are:
(L=6) 102342031273, 102342031301, 102342031321, 102342031343,
102342031369, 102342031403
and
(L=7) 63604045061911, 63604045061957, 63604045062013, 63604045062053,
63604045062097, 63604045062149, 63604045062199
There are several sequences with L=6. The first 10 start at
102342031273, 1012835563819, 1070302300183, 2350811300953,
3063433129909, 3104103122173, 3551303300933, 5262316326901,
5426670290957, 6104611400971
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