Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1274 A modified Buss's Conjecture

On June 25, 2026, Davide Rotondo sent a new Conjecture & Puzzle:


This is my last conjecture regarding Buss. References, 1 & 2

Construct the following Table according to the following algorithm:

Start with f(1)=1.
Calculate the smallest prime number obtained by concatenating the smallest number (other than 1) to f(n).
 Calculate R(n) based on the value used in the concatenation to obtain that prime number.
 Calculate f(n+1) = f(n)R(n).


Here are the first ten rows:

f(n) Smallest prime number obtained  by concatenating the smallest number  (other than 1) to f(n) R(n)
1 1(3) 3
3 3(7) 7
21 21(11) 11
231 231(17) 17
3927 3927(23) 23
90321 90321(19) 19
1716099 1716099(13) 13
22309287 22309287(29) 29
646969323 646969323(37) 37
2.3938E+10 23937864951(41) 41


Davide Rotondo conjectures that the third column will display all and only odd prime integers (except 5) withou any repetition.

He has calulated the following values for the third column, R(n):
[3, 7, 11, 17, 23, 19, 13, 29, 37, 41, 67, 53, 71, 47, 97, 109, 113, 107, 31, 151, 59, 73, 127, 43, 131, 101, 137, 157, 103, 227, 149, 181, 223, 193, 211, 241, 167, 251, 79, 83, 163, 347, 197, 173, 293, 263, 367, 307, 257, 463, 383, 239, 179, 419, 571, 577, 283, 89, 373, 823, 379, 233, 541, 929, 617, 619, 631, 709, 673, 1231, 547, 199, 947, 1171, 449, 397, 311, 953, 349, 191, 409, 139, 317, 601, 431, 271, 457, 613, 509, 719, 811, 647, 659, 967]
 
Q1. Can you verify at least the Rotondo's list up to 967

Q2. Can you extend your list and after the end of your list name the smallest prime absent

Q3. Can you prove the Rotondo's conjecture
 "

 



From July 4 to 10, 2026, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantieghem, Michael Branicky, Gennady Gusev, M. F. Hasler, Paul Cleary, Simon Cavegn

***

Giorgos wrote:

Q1. Rotondo's list is correct
Q2. Here are the first 1000 terms:
3,7,11,17,23,19,13,29,37,41,67,53,71,47,97,109,113,107,31,151,59,73,127,43,131,101,137,157,103,227,149,181,223,193,211,241,167,251,79,83,163,347,
197,173,293,263,367,307,257,463,383,239,179,419,571,577,283,89,373,823,379,233,541,929,617,619,631,709,673,1231,547,199,947,1171,449,397,311,
953,349,191,409,139,317,601,431,271,457,613,509,719,811,647,659,967,839,1013,569,439,503,61,1187,433,337,353,1669,421,863,389,487,937,859,
907,887,587,683,661,761,821,1427,1049,521,1181,757,797,607,2729,229,1619,1229,599,523,277,2503,809,857,691,2237,1279,1361,1481,2269,1987,
499,827,1087,977,853,2099,1129,2029,1103,2141,1493,1597,1249,1291,593,641,787,401,881,2423,563,1381,911,739,2411,1601,1657,733,1811,3109,
701,1163,643,313,1327,461,2459,3119,677,1489,1783,991,1823,557,331,1153,479,751,1031,359,1451,1201,2053,1303,1367,1297,769,1931,2011,743,
1531,1151,2131,2239,1301,1429,1613,653,3583,1321,1097,2089,2389,829,1193,1627,1277,2903,1621,491,3697,2351,4813,1093,1019,983,1283,281,
2789,2953,1259,443,1583,1409,2347,269,2687,971,1549,1069,727,1693,1453,1423,2447,2063,1889,1579,4253,1523,3931,2441,2749,2753,3373,1439,
2003,3929,2551,2803,1753,2111,1777,2819,2311,3331,1217,3001,2357,3499,3881,2843,1879,2963,1511,1801,1009,1709,2393,2017,3491,4691,1553,
1741,1637,2707,3527,1399,4079,3083,1061,3257,4679,3803,3907,773,2713,3659,1543,2693,919,1933,3259,2969,4517,1051,4201,3623,2273,877,
1567,2293,1063,1487,3221,4409,1663,1847,3391,3853,5419,3691,1831,4751,3209,3061,1949,2287,2383,5101,6791,2087,5431,1733,1471,5471,5659,
2557,2243,4339,2887,4567,3307,4327,5737,1499,2579,1459,1999,1997,3407,5209,3557,3727,2137,1979,1039,2129,3371,3833,4583,2297,2437,3229,
1571,4231,1759,2081,8461,2671,2521,6473,1289,3433,3911,7723,4259,10709,3011,1213,2741,3863,5003,1559,2767,1901,3613,6427,2083,1877,2153,
2531,2207,3533,5801,2957,2777,4261,1607,4597,4139,7573,1609,2833,1319,6229,4943,6899,3709,2381,2791,1907,3617,8053,1699,5417,4861,1123,
4241,3181,6089,5333,2143,6199,2341,2677,3137,2549,6451,5711,4391,5179,10939,2309,1447,3253,3889,3089,3347,4481,3323,3461,4157,5021,3449,
3079,2221,2971,6263,2179,3469,2591,5503,5231,3559,11827,3457,1789,7103,3571,4297,5273,4519,3701,5449,3823,941,4093,3299,4957,7459,2797,
5531,3169,5519,1091,4639,1117,2281,4931,5087,11617,4621,3643,4733,2621,4013,7757,883,2161,6067,4363,10337,2617,7673,4153,2689,4129,5011,
4999,8951,2339,3947,6833,1787,6619,5563,3917,5857,6217,2333,2251,3121,3037,3607,6763,8219,2377,2039,8329,1307,8311,4951,7883,9739,1373,
7681,3329,1697,6247,4513,5987,4021,1723,3041,3389,4817,2939,6947,3203,10861,6301,7013,2699,7321,6073,4657,2399,5227,6599,6091,5527,5281
,4547,3637,1721,4903,2069,6047,7607,1873,2477,6977,5651,12161,3847,6793,2917,9421,4397,2801,1747,3467,5261,11411,6211,20173,4451,6917,
3581,8761,10529,13001,6563,5557,5237,3821,3943,2027,2663,2467,3517,6637,4637,6271,5689,6967,16217,7703,3301,4289,5807,18301,4073,4523,
4357,5693,2609,4703,6011,6329,4483,4349,12911,3769,4159,5413,6803,6703,1433,4127,9719,16607,17903,12473,6037,4463,7129,8059,3739,
11383,8387,3779,7417,2593,4111,7069,4789,8087,4049,3541,2633,3217,5407,9221,8263,2683,2851,6577,4549,5437,2473,10607,6869,9203,11257,
5099,6469,5591,24691,11437,467,4831,2113,3163,6673,12671,14731,5923,3593,5501,5861,19037,3919,4273,2647,4423,5669,9649,5119,10331,
6143,16253,9833,8599,4993,4787,9209,6359,13093,4783,6551,4507,4051,11069,9127,6029,9067,5233,6173,4019,5009,13709,11813,8423,13159,
7879,15077,10303,1667,3673,2659,7549,1861,3631,9461,10427,3511,6101,5881,5791,7349,12841,3271,2909,8161,9413,7517,4001,5869,6221,
14143,6701,8941,3319,10789,4007,1223,7043,3361,7489,7873,14563,12577,10457,7759,5347,7507,14087,2927,7717,12157,2371,5107,8233,6421,
6823,4889,15073,4271,6317,8803,10687,8753,7727,3923,3761,4099,5647,1913,5153,4721,11213,13591,8537,8641,6911,17093,5081,12073,4871,
6679,13163,7283,4801,8467,13613,2837,23017,4229,19991,7351,13997,5477,11467,10567,11311,4759,5441,4441,5351,6991,4603,9463,10267,
12101,4933,9311,7901,20107,9839,14621,7213,6571,3547,3797,4057,10831,7433,4283,12253,11867,7039,9283,4591,17597,8069,25621,9151,
5821,3677,5023,5147,5897,16187,5521,10133,13649,7211,17959,9403,1867,6343,9473,5851,5323,11681,10103,7937,10979,4003,8117,9029,
7207,3019,12853,3877,10739,7583,4493,8039,8429,9511,6737,8231,14177,11251,10883,15091,8209,5639,11117,10711,8779,11119,5507,2213,
9587,9011,5657,15107,9043,8737,4799,5387,4373,7187,8563,5197,7193,8377,6323,7639,7699,13687,8291,15791,5393,5623,8539,8669,4663,9539,
2897,5077,15061,11087,10487,9697,9491,7297,3191,8663,8821,5981,9613,13003,7951,14939,2203,6287,9787,13879,9859,20323,20117,5779,
21851,16547,23747,5641,10243,8677,7949,6863,10069,4337,8581,11113,7687,17921,33863,5849,6449,16889,9173,2417,8093,15439,9049,16333,
17231,6569,6857,7829,6829,3359,4673,10949,6661,19603,9013,10253,5479,8623,11719,5839,13691,28433,14341,1483,5743,10957,5717,8317,
16529,8929,16493,9721,15287,15551,5303,6607,17579,18181,9161,6581,10141,8431,15859,1237,6079,29137,10987,25717,7109,14713,16651
The smallest prime absent is 997.

Q3. Proving that there will be no repetition is pretty straightforward:
f(n) is the accumulated product of all previously selected primes up to step n.
If p is the next prime with k digits and p was already in the list we would have the concatenation f(n) (p) = f(n)*10^k + p = p*m*10^k + p because p divides f(n) and the concatenation would be
f(n)(p) = p (m*10^k+1) which is composite, so p cannot be in the list in order the concatenation to be prime.
I cannot prove that all primes will appear but I believe it is true.


***
Emmanuel wrote:

I consulted Mathematica's help file on the primality tests that are in use.
They confirmed that the results I found are reliable.
So, I worked a bit further on Puzzle.
I hope you will replace my first answers by the next ones :
__________________________________________

Q1. Confirmed right !

Q2. Extended list :
R = {3,7,11,17,23,19,13,29,37,41,67,53,71,47,97,109,113,107,31,151,59,73,127,43,131,101,137,157,103,227,149,181,223,193,211,241,167,251,79,83,163,
347,197,173,293,263,367,307,257,463,383,239,179,419,571,577,283,89,373,823,379,233,541,929,617,619,631,709,673,1231,547,199,947,1171,449,397,
311,953,349,191,409,139,317,601,431,271,457,613,509,719,811,647,659,967,839,1013,569,439,503,61,1187,433,337,353,1669,421,863,389,487,937,859,
907,887,587,683,661,761,821,1427,1049,521,1181,757,797,607,2729,229,1619,1229,599,523,277,2503,809,857,691,2237,1279,1361,1481,2269,1987,499,
827,1087,977,853,2099,1129,2029,1103,2141,1493,1597,1249,1291,593,641,787,401,881,2423,563,1381,911,739,2411,1601,1657,733,1811,3109,701,1163,
643,313,1327,461,2459,3119,677,1489,1783,991,1823,557,331,1153,479,751,1031,359,1451,1201,2053,1303,1367,1297,769,1931,2011,743,1531,1151,
2131,2239,1301,1429,1613,653,3583,1321,1097,2089,2389,829,1193,1627,1277,2903,1621,491,3697,2351,4813,1093,1019,983,1283,281,2789,2953,1259,
443,1583,1409,2347,269,2687,971,1549,1069,727,1693,1453,1423,2447,2063,1889,1579,4253,1523,3931,2441,2749,2753,3373,1439,2003,3929,2551,2803,
1753,2111,1777,2819,2311,3331,1217,3001,2357,3499,3881,2843,1879,2963,1511,1801,1009,1709,2393,2017,3491,4691,1553,1741,1637,2707,3527,1399,
4079,3083,1061,3257,4679,3803,3907,773,2713,3659,1543,2693,919,1933,3259,2969,4517,1051,4201,3623,2273,877,1567,2293,1063,1487,3221,4409,
1663,1847,3391,3853,5419,3691,1831,4751,3209,3061,1949,2287,2383,5101,6791,2087,5431,1733,1471,5471,5659,2557,2243,4339,2887,4567,3307,4327,
]5737,1499,2579,1459,1999,1997,3407,5209,3557,3727,2137,1979,1039,2129,3371,3833,4583,2297,2437,3229,1571,4231,1759,2081,8461,2671,2521,6473,
1289,3433,3911,7723,4259,10709,3011,1213,2741,3863,5003,1559,2767,1901,3613,6427,2083,1877,2153,2531,2207,3533,5801,2957,2777,4261,1607,4597,
4139,7573,1609,2833,1319,6229,4943,6899,3709,2381,2791,1907,3617,8053,1699,5417,4861,1123,4241,3181,6089,5333,2143,6199,2341,2677,3137,2549,
6451,5711,4391,5179,10939,2309,1447,3253,3889,3089,3347,4481,3323,3461,4157,5021,3449,3079,2221,2971,6263,2179,3469,2591,5503,5231,3559,
11827,3457,1789,7103,3571,4297,5273,4519,3701,5449,3823,941,4093,3299,4957,7459,2797,5531,3169,5519,1091,4639,1117,2281,4931,5087,11617,
4621,3643,4733,2621,4013,7757,883,2161,6067,4363,10337,2617,7673,4153,2689,4129,5011,4999,8951,2339,3947,6833,1787,6619,5563,3917,5857,
6217,2333,2251,3121,3037,3607,6763,8219,2377,2039,8329,1307,8311,4951,7883,9739,1373,7681,3329,1697,6247,4513,5987,4021,1723,3041,3389,
4817,2939,6947,3203,10861,6301,7013,2699,7321,6073,4657,2399,5227,6599,6091,5527,5281,4547,3637,1721,4903,2069,6047,7607,1873,2477,6977,
5651,12161,3847,6793,2917,9421,4397,2801,1747,3467,5261,11411,6211,20173,4451,6917,3581,8761,10529,13001,6563,5557,5237,3821,3943,2027,
2663,2467,3517,6637,4637,6271,5689,6967,16217,7703,3301,4289,5807,18301,4073,4523,4357,5693,2609,4703,6011,6329,4483,4349,12911,3769,
4159,5413,6803,6703,1433,4127,9719,16607,17903,12473,6037,4463,7129,8059,3739,11383,8387,3779,7417,2593,4111,7069,4789,8087,4049,3541,
2633,3217,5407,9221,8263,2683,2851,6577,4549,5437,2473,10607,6869,9203,11257,5099,6469,5591,24691,11437,467,4831,2113,3163,6673,12671,
14731,5923,3593,5501,5861,19037,3919,4273,2647,4423,5669,9649,5119,10331,6143,16253,9833,8599,4993,4787,9209,6359,13093,4783,6551,4507,
4051,11069,9127,6029,9067,5233,6173,4019,5009,13709,11813,8423,13159,7879,15077,10303,1667,3673,2659,7549,1861,3631,9461,10427,3511,
6101,5881,5791,7349,12841,3271,2909,8161,9413,7517,4001,5869,6221,14143,6701,8941,3319,10789,4007,1223,7043,3361,7489,7873,14563,12577,
10457,7759,5347,7507,14087,2927,7717,12157,2371,5107,8233,6421,6823,4889,15073,4271,6317,8803,10687,8753,7727,3923,3761,4099,5647,1913,
5153,4721,11213,13591,8537,8641,6911,17093,5081,12073,4871,6679,13163,7283,4801,8467,13613,2837,23017,4229,19991,7351,13997,5477,11467,
10567,11311,4759,5441,4441,5351,6991,4603,9463,10267,12101,4933,9311,7901,20107,9839,14621,7213,6571,3547,3797,4057,10831,7433,4283,
12253,11867,7039,9283,4591,17597,8069,25621,9151,5821,3677,5023,5147,5897,16187,5521,10133,13649,7211,17959,9403,1867,6343,9473,5851,
5323,11681,10103,7937,10979,4003,8117,9029,7207,3019,12853,3877,10739,7583,4493,8039,8429,9511,6737,8231,14177,11251,10883,15091,8209,5639,
11117,10711,8779,11119,5507,2213,9587,9011,5657,15107,9043,8737,4799,5387,4373,7187,8563,5197,7193,8377,6323,7639,7699,13687,8291,15791,
5393,5623,8539,8669,4663,9539,2897,5077,15061,11087,10487,9697,9491,7297,3191,8663,8821,5981,9613,13003,7951,14939,2203,6287,9787,13879,
9859,20323,20117,5779,21851,16547,23747,5641,10243,8677,7949,6863,10069,4337,8581,11113,7687,17921,33863,5849,6449,16889,9173,2417,
8093,15439,9049,16333,17231,6569,6857,7829,6829,3359,4673,10949,6661,19603,9013,10253,5479,8623,11719,5839,13691,28433,14341,1483,
5743,10957,5717,8317,16529,8929,16493,9721,15287,15551,5303,6607,17579,18181,9161,6581,10141,8431,15859,1237,6079,29137,10987,25717,
7109,14713,16651,3529,15541,8363,13103,4447,8501,16631,14543,10273,12109,5741,4909,9803,10099,9871,12541,6659,3463,8269,10399,8893,
8933,12437,12619,9157,11059,3671,12113,13177,21523,20407,13931,4561,9343,8011,7121,1993,5297,16091,997,18457,15493,8627,9109,8443,
7369,24091,6133,16273,7019,7933,7643,15227,6353,10733,12011,10531,7793,4217,5483,4973,29453,11587,8009,8237,6043,5113,3067,11789,
14293,10799,6871,11717,6389,14029,9377,14639,7057,5171,9661,6379,2861,19597,9041,7621,6361,10499,11471,13291,8089,7559,29311,9419,
7867,7907,17203,10589,15083,11833,13249,9349,9293,15889,5879,8191,9781,29641,5189,22037,10177,21017,11801,27673,15679,11621,
15817,5381,16097,17749,6269,6961,7499,17837,29401,3313,10891,16073,27059,10651,15313,11549,2539,8923,19543} (1134 elements)
The first missing prime is 1021.

I ended my computations as soon as the integer length of f(n) ecceeded 4000. This is its value :
116577193038369......85539 (4004 digits).
According to Mathematica, all encountered primes are proved primes.

Q3.
I cannot prove or disprove the whole conjecture.
But one thing is definitely true :
a prime of R will never be repeated !
This is because f(n) is the product of all the primes that have passed already (you can verify that this is the case with the f(n) I printed a bit higher) ;.
the concatenation of f(n) with one of its prime factors p is clearly divisible by p and thus cannot be prime..
That every prime (different from 5) will appear is quite probable (in my opinion).
But, that there suddenly appears a composite number in R is not impossible. However, such a composite will have only prime factors that are NOT in R and thus
will be "big" ( >= 1042441 = 1021^2 in our case). But f(n) is a number with 4004 digits ... !

***
Michael wrote:

Q1.
Computationally, one can instantly confirm Rotondo's list up to 967 (94 terms).

Q2.
I have computed the list R(n) and S(n) = "the smallest odd prime != 5 that has not appeared in R(1)..R(n)" for n = 1..1320.
At that point, S(1320) = 1021. Indeed, S(1040)..S(1320) = 1021.
See the attached file containing the Table of n, R(n), S(n) for n = 1..1320.

Here is a summary of Records of S(n) and the index i of their eventual appearance as R(i)
7 2
11 3
13 7
29 8
31 19
43 24
61 100
229 127
269 238
467 664
997 1040
1021 [> 1320]

Q3. The conjecture is reasonable since one always starts anew from 3 looking for the next R(n). A proof of this is not possible with current mathematics.

See attached Pu1274-MB.txt

***
Gennady wrote:


Q1. Yes, it is confirmed.
Q2. After calculating 1000 values of the function, it is determined that the smallest missing prime number is 1009.

The full list of 1000 values of function:
[3, 7, 11, 17, 23, 19, 13, 29, 37, 41, 67, 53, 71, 47, 97, 109, 113, 107, 31, 151, 59, 73, 127, 43, 131, 101, 137, 157, 103, 227, 149, 181, 223, 193, 211, 241, 167, 251, 79, 83, 163, 347, 197, 173, 293, 263, 367, 307, 257, 463, 383, 239, 179, 419, 571, 577, 283, 89, 373, 823, 379, 233, 541, 929, 617, 619, 631, 709, 673, 1231, 547, 199, 947, 1171, 449, 397, 311, 953, 349, 191, 409, 139, 317, 601, 431, 271, 457, 613, 509, 719, 811, 647, 659, 967, 839, 1013, 569, 439, 503, 61, 1187, 433, 337, 353, 1669, 421, 863, 389, 487, 937, 859, 907, 887, 587, 683, 661, 761, 821, 1427, 1049, 521, 1181, 757, 797, 607, 2729, 229, 1619, 1229, 599, 523, 277, 2503, 809, 857, 691, 2237, 1279, 1361, 1481, 2269, 1987, 499, 827, 1087, 977, 853, 2099, 1129, 2029, 1103, 2141, 1493, 1597, 1249, 1291, 593, 641, 787, 401, 881, 2423, 563, 1381, 911, 739, 2411, 1601, 1657, 733, 1811, 3109, 701, 1163, 643, 313, 1327, 461, 2459, 3119, 677, 1489, 1783, 991, 1823, 557, 331, 1153, 479, 751, 1031, 359, 1451, 1201, 2053, 1303, 1367, 1297, 769, 1931, 2011, 743, 1531, 1151, 2131, 2239, 1301, 1429, 1613, 653, 3583, 1321, 1097, 2089, 2389, 829, 1193, 1627, 1277, 2903, 1621, 491, 3697, 2351, 4813, 1093, 1019, 983, 1283, 281, 2789, 2953, 1259, 443, 1583, 1409, 2347, 269, 2687, 971, 1549, 1069, 727, 1693, 1453, 1423, 2447, 2063, 1889, 1579, 4253, 1523, 3931, 2441, 2749, 2753, 3373, 1439, 2003, 3929, 2551, 2803, 1753, 2111, 1777, 2819, 2311, 3331, 1217, 3001, 2357, 3499, 3881, 2843, 1879, 2963, 1511, 1801, 1009, 1709, 2393, 2017, 3491, 4691, 1553, 1741, 1637, 2707, 3527, 1399, 4079, 3083, 1061, 3257, 4679, 3803, 3907, 773, 2713, 3659, 1543, 2693, 919, 1933, 3259, 2969, 4517, 1051, 4201, 3623, 2273, 877, 1567, 2293, 1063, 1487, 3221, 4409, 1663, 1847, 3391, 3853, 5419, 3691, 1831, 4751, 3209, 3061, 1949, 2287, 2383, 5101, 6791, 2087, 5431, 1733, 1471, 5471, 5659, 2557, 2243, 4339, 2887, 4567, 3307, 4327, 5737, 1499, 2579, 1459, 1999, 1997, 3407, 5209, 3557, 3727, 2137, 1979, 1039, 2129, 3371, 3833, 4583, 2297, 2437, 3229, 1571, 4231, 1759, 2081, 8461, 2671, 2521, 6473, 1289, 3433, 3911, 7723, 4259, 10709, 3011, 1213, 2741, 3863, 5003, 1559, 2767, 1901, 3613, 6427, 2083, 1877, 2153, 2531, 2207, 3533, 5801, 2957, 2777, 4261, 1607, 4597, 4139, 7573, 1609, 2833, 1319, 6229, 4943, 6899, 3709, 2381, 2791, 1907, 3617, 8053, 1699, 5417, 4861, 1123, 4241, 3181, 6089, 5333, 2143, 6199, 2341, 2677, 3137, 2549, 6451, 5711, 4391, 5179, 10939, 2309, 1447, 3253, 3889, 3089, 3347, 4481, 3323, 3461, 4157, 5021, 3449, 3079, 2221, 2971, 6263, 2179, 3469, 2591, 5503, 5231, 3559, 11827, 3457, 1789, 7103, 3571, 4297, 5273, 4519, 3701, 5449, 3823, 941, 4093, 3299, 4957, 7459, 2797, 5531, 3169, 5519, 1091, 4639, 1117, 2281, 4931, 5087, 11617, 4621, 3643, 4733, 2621, 4013, 7757, 883, 2161, 6067, 4363, 10337, 2617, 7673, 4153, 2689, 4129, 5011, 4999, 8951, 2339, 3947, 6833, 1787, 6619, 5563, 3917, 5857, 6217, 2333, 2251, 3121, 3037, 3607, 6763, 8219, 2377, 2039, 8329, 1307, 8311, 4951, 7883, 9739, 1373, 7681, 3329, 1697, 6247, 4513, 5987, 4021, 1723, 3041, 3389, 4817, 2939, 6947, 3203, 10861, 6301, 7013, 2699, 7321, 6073, 4657, 2399, 5227, 6599, 6091, 5527, 5281, 4547, 3637, 1721, 4903, 2069, 6047, 7607, 1873, 2477, 6977, 5651, 12161, 3847, 6793, 2917, 9421, 4397, 2801, 1747, 3467, 5261, 11411, 6211, 20173, 4451, 6917, 3581, 8761, 10529, 13001, 6563, 5557, 5237, 3821, 3943, 2027, 2663, 2467, 3517, 6637, 4637, 6271, 5689, 6967, 16217, 7703, 3301, 4289, 5807, 18301, 4073, 4523, 4357, 5693, 2609, 4703, 6011, 6329, 4483, 4349, 12911, 3769, 4159, 5413, 6803, 6703, 1433, 4127, 9719, 16607, 17903, 12473, 6037, 4463, 7129, 8059, 3739, 11383, 8387, 3779, 7417, 2593, 4111, 7069, 4789, 8087, 4049, 3541, 2633, 3217, 5407, 9221, 8263, 2683, 2851, 6577, 4549, 5437, 2473, 10607, 6869, 9203, 11257, 5099, 6469, 5591, 24691, 11437, 467, 4831, 2113, 3163, 6673, 12671, 14731, 5923, 3593, 5501, 5861, 19037, 3919, 4273, 2647, 4423, 5669, 9649, 5119, 10331, 6143, 16253, 9833, 8599, 4993, 4787, 9209, 6359, 13093, 4783, 6551, 4507, 4051, 11069, 9127, 6029, 9067, 5233, 6173, 4019, 5009, 13709, 11813, 8423, 13159, 7879, 15077, 10303, 1667, 3673, 2659, 7549, 1861, 3631, 9461, 10427, 3511, 6101, 5881, 5791, 7349, 12841, 3271, 2909, 8161, 9413, 7517, 4001, 5869, 6221, 14143, 6701, 8941, 3319, 10789, 4007, 1223, 7043, 3361, 7489, 7873, 14563, 12577, 10457, 7759, 5347, 7507, 14087, 2927, 7717, 12157, 2371, 5107, 8233, 6421, 6823, 4889, 15073, 4271, 6317, 8803, 10687, 8753, 7727, 3923, 3761, 4099, 5647, 1913, 5153, 4721, 11213, 13591, 8537, 8641, 6911, 17093, 5081, 12073, 4871, 6679, 13163, 7283, 4801, 8467, 13613, 2837, 23017, 4229, 19991, 7351, 13997, 5477, 11467, 10567, 11311, 4759, 5441, 4441, 5351, 6991, 4603, 9463, 10267, 12101, 4933, 9311, 7901, 20107, 9839, 14621, 7213, 6571, 3547, 3797, 4057, 10831, 7433, 4283, 12253, 11867, 7039, 9283, 4591, 17597, 8069, 25621, 9151, 5821, 3677, 5023, 5147, 5897, 16187, 5521, 10133, 13649, 7211, 17959, 9403, 1867, 6343, 9473, 5851, 5323, 11681, 10103, 7937, 10979, 4003, 8117, 9029, 7207, 3019, 12853, 3877, 10739, 7583, 4493, 8039, 8429, 9511, 6737, 8231, 14177, 11251, 10883, 15091, 8209, 5639, 11117, 10711, 8779, 11119, 5507, 2213, 9587, 9011, 5657, 15107, 9043, 8737, 4799, 5387, 4373, 7187, 8563, 5197, 7193, 8377, 6323, 7639, 7699, 13687, 8291, 15791, 5393, 5623, 8539, 8669, 4663, 9539, 2897, 5077, 15061, 11087, 10487, 9697, 9491, 7297, 3191, 8663, 8821, 5981, 9613, 13003, 7951, 14939, 2203, 6287, 9787, 13879, 9859, 20323, 20117, 5779, 21851, 16547, 23747, 5641, 10243, 8677, 7949, 6863, 10069, 4337, 8581, 11113, 7687, 17921, 33863, 5849, 6449, 16889, 9173, 2417, 8093, 15439, 9049, 16333, 17231, 6569, 6857, 7829, 6829, 3359, 4673, 10949, 6661, 19603, 9013, 10253, 5479, 8623, 11719, 5839, 13691, 28433, 14341, 1483, 5743, 10957, 5717, 8317, 16529, 8929, 16493, 9721, 15287, 15551, 5303, 6607, 17579, 18181, 9161, 6581, 10141, 8431, 15859, 1237, 6079, 29137, 10987, 25717, 7109, 14713, 16651]


***
Hasler wrote:

I just discover https://www.primepuzzles.net/puzzles/puzz_1274.htm
I don't know whether you have followed the discussion on the SeqFan mailing list.

I have explained why concatenation(f,x) = f * 10^k + x is essentially equivalent
to simply adding x to f, regarding divisibility :
p | (f+x) <=> p | (f*10^k + x) except for p = 2 or 5
which therefore are excluded in the "concatenation" version.
(Also, requiring concat(f,x) to be prime excludes *a priori* a solution with x ending in 0,2,4,5,6,8, since a prime can't end in an even digit or 5.
In Buss' variant an x ending in 5 is excluded only after 5 and earlier 2 have occurred as solution, which implies that all future f will end with 0.
So, larger numbers ending in 5 are also excluded, but only due to the fact that 2 and 5 did occur, not a priori, as in the Rotondo variant,
(where concatenation "implies multiplication of f by 10").

It was me who sent the list of terms up to R(94) = 967 to the SeqFan list,
with PARI/GP code to compute more terms if you wish.
The numbers f(n) become large, e.g., f(94) has already 212 digits,
and f(150) has 375 digits, but for PARI this is not a problem.
The list up to 200 goes on
R(95,...,200) = [839, 1013, 569, 439, 503, 61, 1187, 433, 337, 353, 1669, 421, 863, 389, 487, 937, 859, 907, 887, 587, 683, 661, 761, 821, 1427, 1049, 521, 1181, 757, 797, 607, 2729, 229, 1619, 1229, 599, 523, 277, 2503, 809, 857, 691, 2237, 1279, 1361, 1481, 2269, 1987, 499, 827, 1087, 977, 853, 2099, 1129, 2029, 1103, 2141, 1493, 1597, 1249, 1291, 593, 641, 787, 401, 881, 2423, 563, 1381, 911, 739, 2411, 1601, 1657, 733, 1811, 3109, 701, 1163, 643, 313, 1327, 461, 2459, 3119, 677, 1489, 1783, 991, 1823, 557, 331, 1153, 479, 751, 1031, 359, 1451, 1201, 2053, 1303, 1367, 1297, 769, 1931]

Here, f(200) has 527 digits.
The smallest primes that didn't occur up to there are {2, 5, 269, 281, 443, 467, 491, 653, 727,...}

I can't prove Rotondo's conjecture, but again, the problem is exactly the same as for Buss' conjecture (which is also unproven).


***
Paul wrote:

Q1. Done, I get the same.

Q2. I ran the list up to 300 terms.

R(n) list = {3,7,11,17,23,19,13,29,37,41,67,53,71,47,97,109,113,107,31,151,59,73,127,43,131,101,137,157,103,227,149,181,223,193,211,241,167,251,79,83,
163,347,197,173,293,263,367,307,257,463,383,239,179,419,571,577,283,89,373,823,379,233,541,929,617,619,631,709,673,1231,547,199,947,1171,449,397,311,
953,349,191,409,139,317,601,431,271,457,613,509,719,811,647,659,967,839,1013,569,439,503,61,1187,433,337,353,1669,421,863,389,487,937,859,907,887,587,
683,661,761,821,1427,1049,521,1181,757,797,607,2729,229,1619,1229,599,523,277,2503,809,857,691,2237,1279,1361,1481,2269,1987,499,827,1087,977,853,
2099,1129,2029,1103,2141,1493,1597,1249,1291,593,641,787,401,881,2423,563,1381,911,739,2411,1601,1657,733,1811,3109,701,1163,643,313,1327,461,2459,
3119,677,1489,1783,991,1823,557,331,1153,479,751,1031,359,1451,1201,2053,1303,1367,1297,769,1931,2011,743,1531,1151,2131,2239,1301,1429,1613,653,
3583,1321,1097,2089,2389,829,1193,1627,1277,2903,1621,491,3697,2351,4813,1093,1019,983,1283,281,2789,2953,1259,443,1583,1409,2347,269,2687,971,
1549,1069,727,1693,1453,1423,2447,2063,1889,1579,4253,1523,3931,2441,2749,2753,3373,1439,2003,3929,2551,2803,1753,2111,1777,2819,2311,3331,1217,
3001,2357,3499,3881,2843,1879,2963,1511,1801,1009,1709,2393,2017,3491,4691,1553,1741,1637,2707,3527,1399,4079,3083,1061,3257,4679,3803,3907,773,2713,3659}

***
Simon wrote:

Q1. Verified.
Q2.
3, 7, 11, 17, 23, 19, 13, 29, 37, 41, 67, 53, 71, 47, 97, 109, 113, 107, 31, 151, 59, 73, 127, 43, 131, 101, 137, 157, 103, 227, 149, 181, 223, 193, 211, 241, 167, 251, 79, 83, 163, 347, 197, 173, 293, 263, 367, 307, 257, 463, 383, 239, 179, 419, 571, 577, 283, 89, 373, 823, 379, 233, 541, 929, 617, 619, 631, 709, 673, 1231, 547, 199, 947, 1171, 449, 397, 311, 953, 349, 191, 409, 139, 317, 601, 431, 271, 457, 613, 509, 719, 811, 647, 659, 967, 839, 1013, 569, 439, 503, 61, 1187, 433, 337, 353, 1669, 421, 863, 389, 487, 937, 859, 907,..., 26539, 9431, 35729, 33721, 46229, 30809, 14551, 61979, 52249, 31573 (2425 total terms)

First missing prime: 1021

***

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