A 5-digit base-N prime number P is such that we will obtain three other prime numbers by changing its first digit.
Determine the minimum value of N.
clearvars,clc
for N=5:36
p=nextprime(N^4);
pb=dec2base(p,N);
ct=0;
for fd=1:N-1
pb(1)=dec2base(fd,N);
p=base2dec(pb,N);
if isprime(p)
ct=ct+1;
end
end
if ct>3
fprintf('%3d %3d\n',N,ct)
for fd=1:N-1
pb(1)=dec2base(fd,N);
p=base2dec(pb,N);
if isprime(p)
fprintf('%3d %6s %8d\n',N,pb,p)
end
end
disp(' ')
end
end
Finds that 6 is the minimum value for N. Starting with 10001 (decimal 1297), the extra three are 20001, 30001 and 50001, seen below with their decimal equivalents.
Here's the list of all cases through N = 36, the limit of the ability to express such numbers succinctly using conventional digits and alphabetic digits.
Base 12 even has a set of 5 (original plus 4 variants), and base 24 has the maximum for this limited set, 7 (i.e., 1 + 6).
6 4
6 10001 1297
6 20001 2593
6 30001 3889
6 50001 6481
12 5
12 10007 20743
12 20007 41479
12 50007 103687
12 A0007 207367
12 B0007 228103
14 4
14 10011 38431
14 20011 76847
14 70011 268927
14 D0011 499423
15 4
15 10002 50627
15 70002 354377
15 90002 455627
15 D0002 658127
20 4
20 10001 160001
20 70001 1120001
20 C0001 1920001
20 F0001 2400001
21 4
21 10002 194483
21 50002 972407
21 H0002 3306179
21 J0002 3695141
24 7
24 10001 331777
24 30001 995329
24 60001 1990657
24 80001 2654209
24 G0001 5308417
24 K0001 6635521
24 N0001 7630849
27 4
27 1000G 531457
27 3000G 1594339
27 7000G 3720103
27 H000G 9034513
28 4
28 10001 614657
28 C0001 7375873
28 F0001 9219841
28 G0001 9834497
30 6
30 1000D 810013
30 2000D 1620013
30 3000D 2430013
30 6000D 4860013
30 K000D 16200013
30 M000D 17820013
32 5
32 10007 1048583
32 90007 9437191
32 A0007 10485767
32 C0007 12582919
32 U0007 31457287
33 6
33 10008 1185929
33 50008 5929613
33 B0008 13045139
33 D0008 15416981
33 L0008 24904349
33 T0008 34391717
36 5
36 1000B 1679627
36 2000B 3359243
36 D000B 21835019
36 L000B 35271947
36 N000B 38631179
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Posted by Charlie
on 2023-07-17 09:58:15 |
computer solution
