Problems & Puzzles: Puzzles

 Puzzle 1017. Sum of reciprocal of primes such that... Metin Sariyar proposes the following puzzle: I was curious  if there is a prime p=abcdef such that:   1/2 + 1/3 + 1/5 + 1/7 + 1/11 +  + 1/p =a.bcdef . or =ab.cdef...etc.   And I found the smallest example of this: 1/2 + 1/3 + 1/5 + 1/7 + 1/11 +  + 1/2803 = 2.803   The primes I found with this property are: 2803, 31183, 336643, ...? Last minute update: The results sent by Metin are wrong. Please go ahead with the puzzle but forget them. Q1: Can you find the 4-th prime or more primes with this property? Q2: Can you find a palindromic prime with this property where all denominators are also palindromic primes (1/2+1/3+...+1/11+1/101...)?

During the week 12-18 Set. 2020, contributions came from Giorgos Kalogeropoulos, Adam Stinchcombe, Simion Cavegn, Emmanuel Vantieghem, Oscar Volpatti.

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Giorgos wrote:

Q1.
The first two correct values are 13 and 2311

1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 =  1.344022644...

1/2 + 1/3 + 1/5 + 1/7 + 1/11 +  + 1/2311 = 2.311468780....

No other values found up to the prime p=31094741

Q2.
No palindromic prime found with this property after checking the first 16877 palindromic prime numbers
(16877 is the number of palindromic numbers in the first billion prime numbers)

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I got up to p=1208269 before my system decided I was using too many decimal digits, and the only ones I found were 13 and 193 and 2311:

1/ 2 +  + 1/13 = 1.344022644
1/ 2 +  + 1/193 = 1.938932807
1/ 2 +  + 1/2311 = 2.311468780

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Simon wrote:

Q1: Found the first 3 solutions:
sum:1.34402264402264, p:13
sum:1.93893280716845, p:193
sum:2.31146878036571, p:2311

Searched up to sum:3.4505064770485898132979098046, p:34505064791

I tried to calculate the expected number of solutions for the following ranges:
10^10 to 10^30: Expected number of solutions: ~0.5
10^10 to 10^300: Expected number of solutions: ~1.5
10^10 to 10^3000: Expected number of solutions: ~2.5
10^10 to 10^30000: Expected number of solutions: ~3.5
=> I think there are infinitely many solutions for Q1.

Q2: Could not find any solutions.

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Emmanuel wrote:

Q1
p               Sum(1/p)
13         1.344022644022644022...
193       1.938932807168450785...
2311     2.311468780365707911...

The next  p  will be greater than 3*10^8

I found no palprime solution for  Q2  less than  10^15

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Oscar wrote:

About Q1I found the first three correct solutions:
1/2+...+1/13 = 1.344022644
...
1/2+...+1/193 = 1.938932807
...
1/2+...+1/2311 = 2.311468780
...
but no more solutions with p<=10^13.

About Q2, I found no solutions at all with p<=10^21.

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