M F Hasler <mhasler@dsi972.fr> wrote:

Except for the fact that there was certainly no month more that 10^20 (or so) years ago, because the universe didn't exist yet and/or more specifically, time as we define it doesn't reach that far into the past.

Yes, as I mentioned. The Big Bang was a mere 10^10.14 years ago. Whether time existed before that, nobody knows. But this is math, not physics or history. Numbers go on forever. Both Earth and the known universe were around 10^12 days ago, but not 10^13. 33.0% of all days in a Gregorian period (400 years or 146097 days) are the 10^Nth day of some month in the past 10^12 days. That's just under a third. The Gregorian calendar was in use 10^5 days ago (albeit only in Catholic countries), but not 10^6. 18.8% of all days in a Gregorian period are the 10^Nth day of some month in the past 10^5 days.

But yes, the answer to the original question relies only on residues (mod 7) of powers of 10 , so it is essentially the same for any calender that has weeks of seven days and no exceptions to that. It is quite obvious that one particular value will never appear as such a residue,

Right, since 10 and 7 are relatively prime.

slightly less obvious that all others will (and do so for N < 7 ? incidentally corresponding to roughly the age of mankind).

Only because the decimal expansion of 1/7 repeats after 6 digits. Had I tried a different base, the expansion could repeat after any of 1, 2, 3, or 6 digits. If not 6, then several days of the week would never appear. And it's not obvious to me that if I were to re-run my second question for other bases that I'd get a solution for every day, e.g. that every day is the 11^Nth day of some prior month (possibly in the absurdly distant past) for some N. Here's how many of the 146097 days in a Gregorian period are the 10^Nth day of some month for each N, not counting days for which a smaller N applies. There are 4800 for N=0 since every month has a first day, and there's obviously never any previous month with a smaller N. There are 4800 for N=1 since every month has a tenth day, which is of course never also its first day. But there are only 4097 for N=2 since some 100th days of a month land on the 10th day of subsequent month. The smallest N that doesn't appear is 75. The largest N that does appear is 306, for just 9 out of every 146097 days. The largest N with 4800 is 9; the billionth day of each month somehow manages to never land on any 1st, 10th, 100th, 1000th, ... or hundred millionth day of any month. 4800 0 4800 1 4097 2 4800 3 4800 4 4198 5 3234 6 3247 7 4466 8 4800 9 1316 10 2322 11 1354 12 4264 13 4632 14 2834 15 2906 16 3565 17 3246 18 2807 19 2169 20 1768 21 1881 22 2949 23 3752 24 799 25 551 26 259 27 4466 28 989 29 1896 30 753 31 1861 32 1144 33 1827 34 2505 35 1505 36 1579 37 1482 38 2345 39 982 40 721 41 1882 42 856 43 1440 44 1622 45 1546 46 560 47 1641 48 843 49 405 50 1169 51 975 52 619 53 714 54 1099 55 1003 56 460 57 137 58 777 59 257 60 1149 61 905 62 748 63 287 64 124 65 39 66 632 67 74 68 397 69 1222 70 58 71 141 72 341 73 620 74 181 76 110 77 153 78 10 79 446 80 346 81 647 82 371 83 63 84 258 85 173 86 175 87 243 88 243 89 44 90 44 91 17 92 12 93 195 94 22 95 101 96 539 97 219 98 87 99 93 100 134 101 94 102 266 103 304 104 173 105 107 106 376 107 404 108 22 109 102 110 77 111 280 112 37 114 58 115 121 116 12 118 187 119 66 120 17 121 44 122 35 123 240 124 106 125 80 126 121 127 16 128 114 130 114 131 99 132 12 133 6 134 21 135 18 136 140 137 14 138 411 140 87 141 9 142 9 143 54 145 33 146 48 147 20 148 28 149 15 150 5 151 10 152 7 154 2 155 6 156 307 159 2 161 189 162 7 163 12 164 15 165 48 166 10 169 9 170 9 172 81 176 82 177 23 178 25 181 2 182 28 183 26 184 9 186 5 187 16 188 33 190 17 197 16 199 4 202 2 204 35 206 23 213 12 216 17 219 3 222 2 223 13 225 33 227 3 229 1 238 1 261 2 280 9 306 A hundred days ago was last October. A thousand days ago was in April of 2018. Ten thousand days ago was in September 1993. A hundred thousand days ago was in April 1747, less than six years before Britain and its colonies adopted the Gregorian calendar. A million days ago was in February of 718 BCE, 35 years after Rome's founding. Ten million days ago was in the middle of the last glacial period, long before recorded history. A hundred million days ago was early in the third from last glacial period. A billion days ago was during the Pliocene epoch, before our species existed. Ten billion days ago was during the Oligocene epoch, early in the age of mammals. A hundred billion days ago was during the Permian period, the last part of the Paleozoic era, before mammals, or even dinosaurs, existed, but after the great coal forests. A trillion days ago was during the late Archean or early Proterozoic eon, when the only life was single cells, and there was little or no oxygen in the atmosphere. Ten trillion days ago was twice as long ago as the Big Bang. But there are isotopes that will last much longer than that, stars that will still be shining longer than that, and computer programs that would take much longer than that to finish.