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?

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

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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)

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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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