Problems & Puzzles: Puzzles

Puzzle 744. The smallest prime such that...

Here we ask you to get the smallest prime number that uses all the possible distinct letters in your native language when you name that prime number as a cardinal number.

Of course you are able to use any letter more than once, but the count of the letters is only for the distinct ones.

For example, in the Spanish alphabet there are 27 distinct letters but the following six are never used in any number: f, j, k, ń, w, x; accordingly in this language you must produce the smallest prime number whose name uses 21 distinct letters,

Q1. Send the asked prime
Q2. Send also the smallest integer for the same conditions.


Contributions came from Giovanni Resta, Claudio Meller, Emmanuel Vantieghem and Hakan Summakoglu.

Remark: I wrote incorrectly that in Spanish "six (letters) are never used in any number: f, j, k, ń, w, x". This is false because "x" is used in the Spanish word "sextillón". In this same statement I'm supposing that the Spanish word "Vigillón" -that uses the "g" letter- is an accepted term, issue that some people keep under discussion.

***

Giovanni wrote:

In Italian we use 16 letters (plus the 'z' which can appear only in "zero" = 0), namely :
a, b, c, d, e, i, l, m, n, o, q, r, s, t, u, v.

The letter "b" is contained only in "bilione" and "biliardo"
which differ from the English billion and billiard since their
values are 10^12 and 10^15, respectively.

The smallest prime containing all the characters is
un bilione quattromila settecento diciannove = 1000000004719

and the smallest integer should be
un bilione duemila quattrocento ventisei = 1000000002426

***

Claudio wrote:

En espańol estos son primos los que encontré:

100000100000000000000000000000100003910583
Un septillon un sextillon un billon tres millones  novecientos diez mil quinientos ochenta y tres
21 letras diferentes

100000000000000000000000000000000000000000000000000000000000000000000
0000000001000001000000000000000000000001000005410587
Un vigillon un septillon un sextillon un billon cinco millones  cuatrocientos diez mil quinientos ochenta y siete
22 letras diferentes

No sé si son los menores.

***

Emmanuel wrote:

In Dutch, 24 letters are used in the names of the numbers : a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,v,w,x,z (as far as I know  k  and  y are not used).
Perhaps the smallest number that uses all these letters is : 1000001000000000001000000000001000001002568
          (een septiljoen en een sextiljoen en een quadriljoen en een biljoen en een miljoen tweeduizend vijfhonderd acht en zestig).
The smallest prime : 1000001000000000001000000000001000001025869
          (een septiljoen en een sextiljoen en een quadriljoen en een biljoen en een miljoen vijf en twintigduizend achthonderd negen en zestig)

***

 

Hakan wrote:

In Turkish alphabet there are 29 letters,

Small scale:

8 letters (c, f, g, ğ, h, j, p, v) never used in any number <10^12.

Smallest prime containing 21 different letters is 168433: Yüz altmış sekiz bin dört yüz otuz üç.
 

Large scale:

5 letters (c, f, ğ, h, j) never used in any number.

Smallest prime containing 24 different letters is

1000000000000000000000000000000000000001000000000000000000034263:

Bir vigintilyon bir septilyon otuz dört bin iki yüz altmış üç.

In English alphabet there are 26 letters,

Small scale:

6 letters (c, j, k, p, q, z) never used in any number <10^12.

Smallest prime containing 20 different letters is 1001024867: One billion one million twenty four thousand eight hundred sixty seven.

Large scale:

3 letters (j, k, z) never used in any number.

Smallest prime containing 23 different letters is 1001000000001000001001086523:

One octillion one septillion one quadrillion one billion one million eighty six thousand five hundred twenty three.

***

Carlos Rivera wrote on (June 30, 2014)

I must to say that regarding to the following numbers obtained by me, what I have done is just to analyze the results obtained by Claudio Meller (see his primes above) and minimize them, so I would say that these should be regarded as records in collaboration between he and me:

a) A smaller prime number in Spanish with 22 letters, than the found by Claudio Meller

1000000000000000000000000000000000000000000000000000000000000000000000000000001000001
000000000000000000000001000003710587 (10^120+10^42+10^36+10^12+3710587) is a prime number.

UN VIGiLiÓn un SEPTillón un seXtillón un Billón tRes Millones  seteCientos DieZ mil Quinientos ocHentA Y siete (least prime number 22 letters Spanish)

b) Accordingly the least integer with 22 letters in Spanish would be:

1000000000000000000000000000000000000000000000000000000000000000000000000000001000001
000000000000000000000001000000010583 (10^120+10^42+10^36+10^12+10583)

UN VIG
iLiÓn un SEPTillón un seXtillón un Billón DieZ Mil Quinientos oCHentA Y tRes (least natural number 22 letters Spanish)

c) I have unable to produce a smaller prime number using 21 letters in Spanish than the already produced by Claudio.

d) Regarding the least integer with 21 letters in Spanish would be:

10000010000000000000000000000010000310589 (10^42+10^36+10^12+310589)
UN SEPTILlÓn un seXtillon un Billón tReCientos DieZ Mil Quinientos ocHentA Y nueVe (least natural number 21 letters Spanish)

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