## Tuesday, July 30, 2024

### Fun with digital roots

Éric Angelini's most recent blog is here. I was sufficiently intrigued with his challenge to find larger primes than his 20248751248751248751248751248751248751248751248751 that I gave it a go:

z(41) = 20 grows into z(89), Éric's 50-digit prime.
z(91) = 22 grows into z(280), a 191-digit prime.
z(300) = 35 grows into z(1064), a 766-digit prime.
z(3740) = 238 grows into z(d+3737), a d-digit (>78000) prime.

Up to this point, we have 151 primes in Z. Their positions/indices are: 2, 3, 4, 20, 21, 23, 26, 29, 31, 34, 37, 38, 40, 89, 280, 281, 284, 287, 290, 291, 293, 296, 299, 1064, 1066, 1073, 1078, 1079, 1081, 1084, 1085, 1144, 1147, 1170, 1171, 1184, 1221, 1262, 1263, 1265, 1268, 1271, 1278, 1280, 1287, 1616, 1617, 1619, 1660, 1665, 1698, 1700, 1703, 1706, 1707, 1712, 1719, 1721, 1724, 1729, 1784, 1787, 1789, 1792, 1897, 1899, 1914, 1919, 1920, 1922, 1965, 1972, 1973, 1978, 1983, 1986, 1993, 1998, 2001, 2022, 2043, 2045, 2064, 2075, 2076, 2097, 2100, 2103, 2104, 2106, 2109, 2112, 2115, 2225, 2242, 2243, 2245, 2248, 2293, 2336, 2338, 2363, 2366, 2369, 2371, 2388, 2393, 2396, 2513, 2516, 2517, 2660, 2693, 2696, 2697, 2699, 2704, 2709, 2724, 2727, 2744, 2747, 2748, 2750, 2795, 2800, 2801, 2803, 2810, 3156, 3183, 3188, 3191, 3204, 3206, 3225, 3290, 3293, 3360, 3427, 3462, 3475, 3477, 3484, 3485, 3487, 3506, 3511, 3512, 3515, 3738. So we are looking for prime #152 (the next one). I've put an indexed file of Z (to 3752) here.

One might suppose that in the absence of a definite d, we cannot continue Z. Actually, assuming that d exists, we can. What we cannot do is assign indices to the continuation, unless one is ok with:

d+3738  239  (prime #153)
d+3739  240
d+3740  241  (prime #154)
d+3741  242
d+3742  2428
d+3743  24287
...

I've computed 11288 terms of this continuation but, in order to make the file size smaller, I have removed d+7693 to d+15024 from view. The file is here. The final term at the bottom (index d+15025) is prime #190, a 7348-digit prime.

#### 1 comment:

1. Merci beaucoup Hans, as usual!