Problems & Puzzles: Puzzles

Some days ago Jean-Claude Rosa (*) Math teacher of a College in Cluny, France, sent the following email:

"I would like to find a prime number responding to those two conditions:

- this prime number is part of a prime quadruplet
- this prime number also brings forth a prime decade.

I found one of those numbers : 19.

- 19 is part of {11 ; 13 ; 17 ; 19}
- 19 brings forth {191 ; 193 ; 197 ; 199}

Could you help me to find another one ?"

With the help of a little code in Ubasic I found the next earlier example:

D1: {347981, 347983, 347987, 347989 }
D2: {3479891, 3479893, 3479897, 3479899}

Questions:

1. Can any one of the primes of the first decade (D1) be the generator of the next prime decade (D2)?

2. Is it possible to generate a prime decade (D1) that generates - by recursion of the same procedure - two prime decades more (D2, D3 such that , D1 -> D2 -> D3)? If yes, say how exactly and find one example.

Chris Nash sent (15/7/01) the following comments:

"I have done some quick calculations on Puzzle 140 - I do not think I will search much further, since it is going to tackle a long time to find a successful solution (it requires to search until 12 numbers are all primes). But here are my 'partial' answers.

1) Let D1 is the set of numbers 10x+1 10x+3 10x+7 10x+9. Then D2 is the set of numbers 100x+10q+1 100x+10q+3 100x+10q+7 100x+10q +9 where q is 1, 3, 7, 9, depending on which prime in D1 we chose to give D2.

If q=1 or 7, consider the 8 values modulo 3. In particular 10x+1 = x+1 mod 3, 10x+9 = x mod 3, 100x+10q+1 = x+2 mod 3. Since all possible values mod 3 are covered, at least one of these values is divisible by 3. Since none of them are equal to 3, this set of values must contain a composite.

q=9 you have shown a result.

For q=3, there is no reason why a solution should not exist. But tests indicate it is much larger than the q=9 solution (no solution for x<2^30). In both cases, x must equal 1 modulo 3 for solutions to exist.

1000x+100q+ 10r+1
1000x+100q+ 10r+3
1000x+100q+ 10r+7
1000x+100q+ 10r+9

From the first part q and r must both be either 3 or 9. It already appeared that 3 made it harder to find solutions, and in fact if both q =3 and r=3, there are no solutions to this second problem - at least one of the 12 numbers must be divisible by 7.

In the remaining cases, there is only one valid value of x mod 3 as before, and also only one valid value of x mod 7, so it should be possible to accelerate the search using that. In any case however there is no reason why a solution should not exist, although it may take some time to find. I am going to move my code to extended-precision to search a little further..."

***

Jean Claude Rosa has sent (13/9/01) three "shortcuts" to find a solution to question 2):

Let D1=p1,p2,p3,p4 > D2=q1,q2,q3,q4 > D3=r1,r2,r3,r4 D4=s1,s2,s3, s4 > The search can be drive on three ways:

Way 1: p1=210*k+11;q1=2100*k+191;r1=21000*k+1991;s1=210000*k+19931

Way 2: p1=210*k+101;q1=2100*k+1031;r1=21000*k+10391;s1=210000*k+103931

Way 3: > p1=210*k+191;q1=2100*k+1991;r1=21000*k+19931;s1=210000*k+199391

***

J. Wroblewski wrote (1/7/6):

I have got some new results regarding your

List of smallest numbers of the form X1, such that
X1, X3, X7, X9, X91, X93, X97, X99
are all prime (X stands for any sequence of decimal digits):

11
347981
43265681
156552491
159404891
162785471
176470961
190768211
209164601
321630311
338404481
387135641
527311061
536434511
568441871
738653561
746688791
964496381
1201909601
1280488211
1340711621
1670295701
1985742371
2466094451
2711397971
3358787891
3558992621
3950245961
4362432071
4419340571
4586936741
4596569441
4797006881
4952521571
4964890751
5777631521
5816477741
5998695161
6117103451

List of smallest numbers of the form X1, such that
X1, X3, X7, X9, X31, X33, X37, X39
are all prime:

40739681
41599001
68820041
96466541
179824361
224526011
249838151
311356181
433234091
628058021
768020081
798732371
878849681
1228175231
1268554031
1271560391
1357188731
1632228881
1845783761
2015668091
2026642481
2183464181
2793387341
2955694451
3319145651
3336314621
3736861691
3890848601
4261734221
4711508441
4802673011
4938483371
5503275221
5747092991
6039941351
6449927501

The least number of the form X1, such that
X1, X3, X7, X9, X91, X93, X97, X99, X991, X993, X997, X999
are all prime:

657835997711

The least number of the form X1, such that
X1, X3, X7, X9, X91, X93, X97, X99, X931, X933, X937, X939
are all prime:

2673748786991

The least number of the form X1, such that
X1, X3, X7, X9, X31, X33, X37, X39, X391, X393, X397, X399
are all prime:

34127252370851

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