Problems & Puzzles: Puzzles

 Puzzle 387. Prime curios in the Pascal's triangle rows? Some time ago I published the following curio: 18285670562881 is the smallest emirp to be formed by concatenating a row of Pascal's triangle. Yesterday my old friend G. L. Honaker, Jr. asked to me: "Could we say, the only known emirp to be formed by..." Let me know. Thanks!" Question: 1. Can you find another emirp in another row? 2. Can you find another prime curio inside the Pascal's triangle rows?

Contributions came from Farideh Firoozbakht. J. K. Andersen discovered that this puzzle was already posed as Puzzle 132. Sorry again...

***

Farideh wrote:

Q1. Between the first 560 numbers formed by concatenating a row of Pascal's
triangle there are only three primes p1, p2 & p3 corresponding to n = 1, 8 & 29.
Also between them there are ony six numbers m  q1, q2 , ..., q5 & q6 such that
reversal(m) is prime corresponding to n = 1, 6, 8, 9, 10 & 32. Next such number
if it exists has more than 67620 digits.
So it seems that there is no other such emirp.

p1 = 11
p2 = 18285670562881
p3 = 129406365423751118755475020156078042921451001500
520030010345972905189593567863915775587607755876
067863915518959353459729020030010100150054292145
1560780475020118755237513654406291   (178 - digit)

q1 = 11
q2 = 1615201561
q3 = 18285670562881
q4 = 193684126126843691
q5 = 1104512021025221012045101
q6 = 13249649603596020137690619233658561051830028048
80064512240129024480225792840347373600471435600
56572272060108039056572272047143560034737360022
57928401290244806451224028048800105183003365856
906192201376359604960496321  (215 - digit)

Q2.

1. Find emirp formed by concatenating some consecutive rows of Pascal's
triangle. For example p(3, 5) = 13311464115101051 is an emirp.

2. Find large primes or probable primes formed by concatenating some
consecutive rows of Pascal's triangle.

Examples:

1. p(36, 46)      (3911 - digit)
2. p(68, 74)      (7484 - digit)

***

One week after Jacques Tramu added:

Tested up to row = 200 (8567  digits).

Found only  3 primes rows  and one emirp :

prime-row      1       11
prime-row       8       18285670562881 (emirp)
prime-row       29      1294063654237511187554750201560780429214510015005200300103459729051895935678639157755876
077558760678639155189593534597290200300101001500542921451560780475020118755237513654406291

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