Problems & Puzzles: Collection 20th

Coll.20th-018. Prime repunits as sum of consecutive primes.

On Set. 10, 2018, Claudio Meller posted the puzzle 1513 in his always interesting site.

In that puzzle you are challenged to find repdigit of all kind as sum of K>1 consecutive primes.

One of the solutions sent by one of the Clauido's readers (Hans Havermann) was this one:

1111111111111111111 (19 digits) = 158730158730158647 + 158730158730158681 + 158730158730158699 + 158730158730158723 + 158730158730158759 + 158730158730158783 + 158730158730158819

I liked this very much!

Accordingly, I dedicate Q1 of this puzzle to find only solutions for the rest of the prime repunits.

Q1. Find a set of K consecutive primes, K minimal, whose sum is equal to a prime repunit, PRU(D, with D digits for D=23, 317, 1031,...

Q2. Find solutions for some other conspicuous prime numbers with D>10 digits (you decide what conspicuous means here...)


Contribution came from Hans Havermann.


Hans wrote on Oct 5, 2018:

I just wrote up my response to Q1:


On Oct 7, 2018, Hans wrote again:

Here's my contribution to Q2:



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