Problems & Puzzles: Puzzles

 Puzzle 438. 1024383257 From the Prime Curios site: 1024383257 You cannot insert a digit in 1024383257 to form another . [Blanchette] Note: 10243832571 is prime, so “insert” is rigorous; it excludes “append” by the extremes.   Q. Can you find a larger prime like this?

Contributions came from Giovanni Resta, Farideh Firoozbakht, J. K Andersen, Jim Howell, Alex Pro & W. Edwin Clark:

The first three of them noticed two ugly things:

a) This puzzle is a kind of repetition of Puzzle 398
b) 10243832571 is not prime

Sorry for both mistakes.

Nevertheless something new was obtained, on my request to GR, FF & JKA:

"...try to get larger emirps which can not come prime by inserting any digit, supposing that the number so formed can be read in both directions" (CR)

JKA wrote:

760685342399 with reversal 993243586067 is the smallest such emirp.

Based on heuristic estimates I definitely expect infinitely many. I have no idea how to prove it, or just how to prove there are infinitely many emirps (looks at least as hard as the twin prime conjecture).
I have searched other variations which are not about backwards reading.

First consider primes satisfying the original puzzle condition: They always become composite if a digit is inserted between the existing digits. The smallest such prime is 97673.
If we also require it's a weakly prime (puzzle 17, it becomes composite if any digit is changed), then the smallest is 530304937.
If we additionally require that it becomes composite if any digit is placed before or after the existing digits (except placing a leading 0), then the smallest is 34101693667.
If we then also require it becomes composite if any digit is removed from the original prime then the smallest is 40144044691.
If we finally require it becomes composite if the value is changed by swapping two digits next to each other (a common typing error), then the smallest is 185113489357.

Note: Many smaller candidates were eliminated by finding probable primes. If some of these were actually composite then there might be smaller solutions.

Farideh wrote:

In fact if b(n)=10^(n-1)+a(n) is the smallest n-digit emirprime which
can not come primes by inserting any digit then a(n) for n=6,7,...,
49 & 50 are:

269293, 453643, 2176657, 12000043, 12898261, 3756771,
1171551, 266929, 19681653, 3896053, 2009079, 21006927,
193063, 8358093, 5835931, 4537903, 10045867, 23212527,
12561981, 14666667, 3407323, 3407323, 14897223, 48580363,
1262089, 10046517, 2397139, 20845107, 4504717, 4494109,
17453671, 11578539, 33440329, 10327309, 9355881, 13721827,
127027939, 2579637, 12344419, 72181917, 37023, 10196281,
1398117, 3351807, 27043537 & 47978253.

So 10^50+47978253 is the smallest 51-digit emirprime which can
not come primes by inserting any digit.

...

a(100)=11180943 so 10^99+11180943 is the smallest 100-digit
emirprime which can not come primes by inserting any digit.

***

Other contributions are:

Farideh wrote:

I found many such primes. p1, p2 & p3 are three of them.

p1=prime(5*10^9 +65845)=122432190211
p2=prime(5*10^9 +84421)=122432663357
p3=prime(5*10^9+106351)=122433220967

Primes p2 & p3 have the further property that we can not add any digit before or after them to form another prime.
So from each of the primes p2 & p3 we get 129 composite numbers.

Alex Pro wrote:

Here are numbers between 1024383277 and 1050000000, which possess wanted property:

1025057647,
1025093873,
(36 more primes)
1049700877

W Edwin Clark wrote:

Here's a 100 digit example

94354184428121374340740165821888656005067256451377673404560109648714031
90575118466172938184982070053

My computer just found this 200 digit example:

99844622247047704293171222218478223002820995931905876929672897090627039
41588309479452351854253139821798222843767094896260153001645045407364005
9041746413937587672432061833797407216280036471348214426941

Giovanni wrote:

just a little note on Puzzle 438.

The smallest palindromic prime with the
requested property is 35082428053.
The only other of the same length is 98688188689

There are exactly 13 such palprimes of 13 digits,
from 1612748472161 to 9741095901479,
and 136 of 15 digits,
from 117163747361711 to 998650222056899.

The smallest one of 17 digits is 10010736763701001.

One may wonder about such primes which have the
property also when expressed in another base.
Here is a table of the minimal primes which
have the property both in base 10 and in base
B for B=2,3,...,9.

B dec. in base B
======================================
2 22038829 1010100000100100100101101
3 710713 1100002220201
4 65561597 3322012033331
5 5828287 2443001122
6 7199644561 3150225211441
7 93717257 2215404104
8 329367011 2350336743
9 179149039 414081247

***

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