Problems & Puzzles: Puzzles

Puzzle 1000. Prove that math* is fun

Arrange this palindromic prime (19973-92401-97879-10429-37991) into a 5-by-5 array and each row, column, and main diagonal will form a reversible prime. [Blanchette]

 1 9 9 7 3 9 2 4 0 1 9 7 8 7 9 1 0 4 2 9 3 7 9 9 1

 19973 92401 97879 10429 37991 19913 92707 94849 70729 31991 12821 30803

This is a nice curio, but... these 12 primes are:

a) Four of them are palindromic: 97879, 94849, 12821 and 30803

b) There are four pairs of reversed primes: (19973, 37991), (92401, 10429), (19913, 31991) and (92707, 70729)

A few hours after the edition of this page, Emmanuel Vantieghem made me notice that if we start with a palprime is inevitable the emergence of these four 5-digits palprimes, and all the other reversed primes...

Q1. Starting with a 25 digits emirp find a solution such that all the 12 five-digits primes are distinct 5 digits emirps.

Q2. Redo Q1 starting with 49 digits emirp, obtaining 16 distinct 7-digits emirps.

Last hour contribution: Oscar Volpatti sent the following generous message by email (May, 7, 2020):

"thank you for your first 999 prime puzzles!

I have a suggestion for the following one:

prove that math is fun. Well, you already proved it, over and over again. Hoping that you'll continue to prove it along 4-digit puzzles."

Q3. Any nice argument, story, joke, cartoon,... and similars, about the statement "math is fun"?

____
* "math" (US), or "maths" (UK)

During the week 9-15, May 2020, contributions came from Giovanni Resta, Emmanuel Vantieghem, Fausto Morales, Carlos Rivera

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Giovanni wrote:

Q1. I found that the smallest and largest emirps of 25
digits that generate a 5x5 square of distinct emirps are:

1119710499900593386399139 and 9979398299703131171937997

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Emmanuel wrote:

Q1. N = 5
If I made no mistake, the smallest 25-integer emirp solution is  1119710499900593386399139.
It defines 12 emirps : 11197, 10499, 90059, 33863, 99139 (rows)
11939, 10039, 14081, 99563, 79939 (columns)
10069, 93097 (diagonals).
Besides, the reverses of these 12 numbers are 12 other primes.
So, we got a total of  24  primes.

It was not asked that the concatenation of the five columns should be an emirp too.
But I could not resist the temptation to see if this supplementary condition could lead to a solution or not.
Finally found this one :
The 25-digit emirp : 1197173681364679139739119 (there is no smaller)
The rows : 11971, 73681, 36467, 91397, 39119
The columns : 17393, 13619, 96431, 78691, 11779
The diagonals : 13499, 18413
All the reverses of these primes are 12 other primes.
The concatenation of the columns is the emirp  1739313619964317869111779.

Q2. N = 7
If I made no mistake, the smallest 49-integer emirp solution is  1111339100003310000371043587726997111709411117973
It defines 16 emirps : 1111339, 1000033, 1000037, 1043587, 7269971, 1170941, 1117973 (rows)
1111711, 1000211, 1004671, 1003907, 3005999, 3338747, 9377113 (columns)
1003943, 9303611 (diagonals).
Besides, the reverses of these 16 primes are 16 other primes.
So, we got a total of  32  primes.

The concatenation of the columns however is not prime.  Neither is its reverse.

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Fausto wrote:

Q3. "There are three kinds of people in the world: those who know math and those who don't.". Neil DeGrasse Tyson, Twitter, 2013.

***

Carlos wrote:

Q3. My contributions for the phrase "math is fun" are:

a) "A mathematician is a machine for turning coffee into theorems". Alfred Reny.

b) "A Mathematician's Apology", G. H. Hardy.

c) Surprise!... "Math is fun" is also a website administrated by Rod Pierce, whose lemma is "We offer mathematics in an enjoyable and easy-to-learn manner, because we believe that mathematics is fun"

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