A prime number of consecutive primes summing to a repdigit prime

I suppose that 2 + 3 = 5 is an example, even though the 5 is a degenerate repdigit number. A more proper solution is 158730158730158647 + 158730158730158681 + 158730158730158699 + 158730158730158723 + 158730158730158759 + 158730158730158783 + 158730158730158819 = 1111111111111111111, that sum being one of the few known repunit primes.

What else?

Murder

More than seven hours after a fatal stabbing on Friday morning, 600 meters northwest of my home, police were still documenting the evidence on the road. We've had more than our share: On Wednesday, 1 km southeast of me, a drive-by shooting killed another man!

I did manage to capture the CP24 news team doing their coverage and — a few minutes later — a nice shot of reporter Arda Zakarian reacting to some police compliments:

More ænlic primes

When I presented ænlic primes earlier this month I pointed to the remarkable stability of the number of ænlic primes for a given digit-length. I noted that — except for four near-the-start values — they are always greater than 20 and less than 48. Last week I finished my search to 10^500, netting a total of 16115 ænlic primes thereto. As a result, the "greater than 20" needs to be adjusted to "greater than 17". Still, the overall trend continued and I decided to spend another week angling for the counts of 660- to 670-digit ænlic primes and — at intervals of ten, as a bridge — from 510- to 650-digit counts:

As icing on the cake, I did the 1000-digit ænlic primes. I found 36:

  (10^1000 - 1)/9 + 7*10^463
2*(10^1000 - 1)/9 - 2*10^647 - 1
2*(10^1000 - 1)/9 - 2*10^192 + 7
2*(10^1000 - 1)/9 + 6*10^239 - 1
2*(10^1000 - 1)/9 + 3*10^251 + 5
2*(10^1000 - 1)/9 +   10^344 + 1
2*(10^1000 - 1)/9 + 4*10^641 + 7
2*(10^1000 - 1)/9 + 6*10^969 + 5
3*(10^1000 - 1)/9 - 2*10^150
3*(10^1000 - 1)/9 + 5*10^481
3*(10^1000 - 1)/9 + 2*10^759
4*(10^1000 - 1)/9 - 4*10^777 + 5
4*(10^1000 - 1)/9 -   10^712 - 1
4*(10^1000 - 1)/9 + 5*10^ 64 + 5
4*(10^1000 - 1)/9 + 4*10^192 + 5
4*(10^1000 - 1)/9 + 4*10^516 + 3
4*(10^1000 - 1)/9 +   10^738 + 3
4*(10^1000 - 1)/9 +   10^796 + 5
4*(10^1000 - 1)/9 + 2*10^930 + 5
5*(10^1000 - 1)/9 - 3*10^812 - 4
5*(10^1000 - 1)/9 - 5*10^485 + 2
5*(10^1000 - 1)/9 - 2*10^ 41 + 2
5*(10^1000 - 1)/9 +   10^611 + 4
6*(10^1000 - 1)/9 - 6*10^799 + 1
6*(10^1000 - 1)/9 - 6*10^461 - 5
6*(10^1000 - 1)/9 - 4*10^170 - 3
6*(10^1000 - 1)/9 - 2*10^161 - 3
6*(10^1000 - 1)/9 + 2*10^284 + 3
6*(10^1000 - 1)/9 + 2*10^891 - 3
7*(10^1000 - 1)/9 - 5*10^517
7*(10^1000 - 1)/9 +   10^538
4*(10^1000 - 1)/9 + 4*10^999 - 3
8*(10^1000 - 1)/9 -   10^401 + 1
8*(10^1000 - 1)/9 - 2*10^337 - 7
8*(10^1000 - 1)/9 - 8*10^125 - 1
8*(10^1000 - 1)/9 - 6*10^ 88 - 1

LCM

Cliff Pickover's tweet from 2015 examples a least common multiple (LCM) of the positive integers to 100. Folk of my generation first learned of LCM in school when adding fractions with differing denominators. Finding the LCM of a range of numbers is fast and easy in Mathematica:

LCM @@ Range[100]
69720375229712477164533808935312303556800

One could look up the answer in OEIS sequence A003418 (its current b-file goes up to 2308). But of course I wanted to know how much further I could take this:

LCM @@ Range[10000]
57933396702876429686922708791662400986348602979985188253931383511489793001457731823088325981761829221665744176794023407056559491402467891577328326763021299466843118474637852656831938521549472347971073068161679301705472685236926463387338495220571064420250677315000599457941340849496227227628926493771018264821842230370349640102573492881424317306189569467101495834601991270039918780924506495405797923762205360790652073159333382795670426041033566699342449050309786673681670483369155689567554239898879039744147333971988258061042090970476729293484513072443614795766878726325795854855394491290821167148355514749149683707585283381546153703014210442470318180511906691108325146494219343498899382918018246586609827667470329166012110874981104800415741527586280026737848182673635645872230905234515169611121042867043956727839314198728626274066655467846183343599194761590368608472578398169740111485924046986870714883894285841394964627408094161019230662749101230783008668676907211199488107523306410531772045452853957706873238466829988649822157557103503283563398281775464911904789159515900987401574678885942493907604740891878907698622679570965569483682456042918236444719794534411171907606336090534029349351300276141892529795448751826394399153216183270385737795748770508612096374765333578237973395907265484337502903901947799663388329849198045756207969590055686607678195206367273600632909417024224754750428711236917913663419215925830944035539848749163178489614227546656090790164108195741048033614368495827231281392190063051315248070192263400801315095608512139510731469732311313898995746040563433121427776071482655904346538281010668476731132415829844984600414136781404774213539507859790229205890271721600309169926806121871750008163738773911610009508609149665332579632767397078877996926581337419351834754370411008686136818501030862345505385357198060894463821342298717851567836562984344806469613768024764967372979655179066074398198246805104576134474823016488842818077041661676098399378809713894284994865370648616800689225595431967181072865363430005250840767890912164530705704936837915584856606960687347372391339254432119085932175541392954343684716695162629271229789289404752104218596977036941910521266321726821940533986384237994403780618301379099347975260122724194454275088825587044488208965690373706904056926509324696308810974331790119456438147168585552011926921912167450509941646104076818762060881903969616431646384985895944231218505620547093874241169759205450145478746112796898626711966320965057212219958567338851356631739947125096250452942497473309299907612330435197454392788637359253116308685007014249605492659524429134513344137517101872279428202285951652856354827230765931502805341696470148698002737700823078904634554776750169178259216255903968865588749827789888950172452455448248833712309835657561369233157977405579365293671943131412034109901944892819245001657496671822581274180596255340507054499934060282320458240722454209933569735940032859109934686878274110864394924463573852015338428881961843292083566034669814619612606638283615766521897504566616272305253931938372830446073384019299355320864342734019517633662346790422915951954822645137091494126100390104510987373366328615363056042137440808225973600809566845718073791616927784260557845021823094999326904373592319407516660896764388092262510369182153559285446074990941863516247226532653142198551840063631989428776799533286215464660644129411503287306838551341184739976807097763115368031748646043780549055143428297230678053738453010234949008253769355207208167999203353157524666017029803679612131824740794652592875662818479980117505768541194835524231818203552256426752730455115752280837099763237606348192867936457993970866446264015812819179994138642295108872381709181937092290392544335464025324661284746003660247161196698209062164637264114930766444473471083408200329662059064201896721165015687487728300854501780810155844837489798144309942999091774466406270065305461848242329380636274754660519867343112275861821293501112101434868225378041813836808745417606289159904294165941408692922250601127804971962342807927743390030395048263275616935165347620718001157478088456439083590834464409622781693790883289597024043982584220069224170235863458745344365684082114430362867446193601075569803650773018026700003812298460527976219100308016537538008597751565631582745643139434508332515569645426771932483266712323523039014220800000

LCM @@ Range[1000000000]
abridged answer

If you want the unabridged answer to the LCM of the positive integers to one billion, it's 440 MB — somewhat unsuitable for web-browser display. If you have a text application that can handle that, I've put a 207 MB .zip compression of it here [clicking on this should download the compressed file]. I discovered an interesting initial-digits convergence when I calculated the number of decimal digits in these to-powers-of-ten LCMs:

Table[Ceiling[Log[10, LCM @@ Range[10^n]]], {n, 9}]
{4,41,433,4349,43452,434115,4342311,43428686,434295176}

The initial 4 is the number of digits in 2520 (LCM to 10) and the next 41 is the number of digits in LCM to 100. The final 434295176 is the number of digits in LCM to one billion.

Size is everything

Harry Allen's tweet yesterday asks about the size comparison of a 2013 Mac Pro to that of a soda can. Of course that Mac Pro looks somewhat like a soda can but it's significantly larger. The volume of the 2013 Mac Pro is roughly 5000 cm^3. The volume of a soda can is typically 400 cm^3. So no, this Mac Pro doesn't qualify as a soda-can-sized computer. Cliff Pickover retweeted the post without comment.

Ænlic primes

Last month I showcased a very large term in OEIS sequence A039986. I didn't then have a special name for these numbers but I've given it some thought and finally settled on ænlic primes. Say what? A program-generated pronunciation [loud] for the Old English expression suggests (to my ear) a tad more ah than eh of that initial ae diphthong. The ending is like the word each. But I digress.

To recap, a term in A039986 is a prime number where no other permutation of its digits is prime — including permutations with leading zeros. For example, using two zero digits and ten one digits we can create the following 66 distinct permutations:

 1    1111111111 = 11*41*271*9091
 2   10111111111 = 101*421*237791
 3   11011111111 = 7*11*13*11000111
 4   11101111111 = 211*52611901
 5   11110111111 = 11*31*43*191*3967
 6   11111011111 = 41*101*271*9901
 7   11111101111 = 11*17*59417653
 8   11111110111 = 19*53*1667*6619
 9   11111111011 = 7*11*13^2*29*29443
10   11111111101 = 23*101*1931*2477
11   11111111110 = 2*5*11*41*271*9091
12  100111111111 = 11*31*293580971
13  101011111111 = 19*101*52637369
14  101101111111 = 7*11*13*17*281*21143
15  101110111111 = 41*269*271*33829
16  101111011111 = 11^2*2791*299401
17  101111101111 = 101*9901*101111
18  101111110111 = 11*59*155795239
19  101111111011 = 89*823*859*1607
20  101111111101 = 7*11*13*41*271*9091
21  101111111110 = 2*5*101*421*237791
22  110011111111 = 11*10181*982321
23  110101111111 = 101*1090110011
24  110110111111 = 7*11*13^2*293*28879
25  110111011111 = 97*659*1722557
26  110111101111 = 11*10010100101
27  110111110111 = 101*149*739*9901
28  110111111011 = 11*109*6047*15187
29  110111111101 = 110111111101
30  110111111110 = 2*5*7*11*13*11000111
31  111001111111 = 11*53*1097*173561
32  111010111111 = 101*181*6072431
33  111011011111 = 7^2*11*13*15842873
34  111011101111 = 383*289846217
35  111011110111 = 11^2*47*19520153
36  111011111011 = 31*101*3581*9901
37  111011111101 = 11*9013*1119707
38  111011111110 = 2*5*211*52611901
39  111100111111 = 11*19*531579479
40  111101011111 = 61*101*18032951
41  111101101111 = 7*11*13*23*71*67967
42  111101110111 = 1373*80918507
43  111101111011 = 11*10100101001
44  111101111101 = 101*241*461*9901
45  111101111110 = 2*5*11*31*43*191*3967
46  111110011111 = 11*41*271*909091
47  111110101111 = 83*101*2887*4591
48  111110110111 = 7*11*13*683*162517
49  111110111011 = 17*71*2777*33149
50  111110111101 = 11^3*83478671
51  111110111110 = 2*5*41*101*271*9901
52  111111001111 = 11*131*77106871
53  111111010111 = 101*5647*194813
54  111111011011 = 7*11*13*199*557789
55  111111011101 = 41*251*271*39841
56  111111011110 = 2*5*11*17*59417653
57  111111100111 = 11*10101009101
58  111111101011 = 101*1100109911
59  111111101101 = 7*11*13*111000101
60  111111101110 = 2*5*19*53*1667*6619
61  111111110011 = 11*1499*6738499
62  111111110101 = 17^2*101*3806609
63  111111110110 = 2*5*7*11*13^2*29*29443
64  111111111001 = 11*307*32902313
65  111111111010 = 2*5*23*101*1931*2477
66  111111111100 = 2^2*5^2*11*41*271*9091

Only #29 (110111111101) is prime and because this number contains all twelve digits (i.e., both zeros) it is an ænlic prime. [There is, for example, only one prime in the 21 permutations of two zeros and five ones but that prime (101111) is missing a zero so it is not ænlic.] If there were more than one prime in any such permutation list, they would not be ænlic. Using PARI code on the OEIS A039986 webpage, I generated 1141 ænlic primes less than 10^30. The program spent two weeks looking for 31-digit solutions — only to abort with a memory issue!

At that point I decided to write my own Mathematica program. Not a foolproof discovery engine that calculated all terms for a given digit-length — because such searches would take too long — but rather, a brute force effort that started with a given number of repdigits (1-9) and replaced up to two of the digits with something else (0-9, but not the repdigit). I can't be certain that this finds all of the ænlic primes that exist for the digit-length under consideration — but I think it probable.

The program is currently up to 10^350. I've preceded the decimal numbers with compact notation (where warranted: all but 44) and indexed the lot. I should be able to reach 10^500 but it will take another couple of weeks and will further deprive my "long compute" of the four cores on my fastest machine. [August 19: done!]

The number of ænlic primes of a given digit-length (4, 13, 34, 45, 68, 67, 47, 36, 40, 46, 33, 45, 35, 38, 32, 39, ...) is remarkably stable:

In my range — except for four near-the-start values — they are always greater than 20 and less than 48. I wonder if there exists a long-term trend.

More ænlic primes

Digit sums of powers of integers

Cliff Pickover's Thursday twitter "shiver" explained: "666 is equal to the sum of the digits of its 47th power." That is really only half the story. 666 is also equal to the sum of the digits of its 51st power. I've created a cheat sheet to easily find exponent solutions for integers up to 20034 (note that not all integers have a solution). With it (and nothing else) you can solve this problem:

The sum of the digits of n^911 is n. What is the sum of the digits of n^913?

(10^10002-1)/9-10^2872

This is what (10^10002-1)/9-10^2872 looks like:

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111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11

A zero (I've made it bold to make it easier to spot) is sandwiched between 7129 ones to the left and 2872 ones to the right. A number composed only of ones is called a repunit, so this would be a near-repunit.

The number is prime of course but that's not what makes it special. It would be relatively easy to come up with other near-repdigit primes of this (or even much larger) size. What makes it special is that no other distinct permutation of its digits is prime.

Think of it this way: Consider the repunit composed of 10002 ones. There are 10002 different integers where one of the ones is replaced with a zero. When the initial one is replaced with a zero, that zero disappears and the integer becomes a repunit composed of 10001 ones. Every other substitution (having the zero in a different place) maintains the 10002-digit length. We test all 10002 thus-permuted integers for primality (which may take a while) and discover that they are all composite except for one, i.e. the provided example.

There is a sequence of such primes (A039986) in the OEIS and I'm currently trying to extend the number of its known initial terms.

A pre-European aboriginal settlement in Weston

In 1937, Frederick David Cruikshank and Joseph Nason published their "History of Weston" in which the authors gave short shrift to the place's aboriginal inhabitants. On page 2 of the first chapter (The Beginnings) they wrote:

"The Humber in this district was a favourite haunt of the Ojibway tribes and it is said that they visited the old hunting ground as late as 1841. There were at least two Indian burial places close to Weston, one just south of the village and the other about a mile to the north. In 1825, John Paul, the well-known cooper at The Humber, saw the internment of an Indian warrior and his accoutrements."

I was curious about that burial place "just south of the village". The context is clearly the original village of Farr's Mills which was on the flood plain of the Etobicoke side of the Humber river, roughly where the Weston Golf and Country Club is now. Just south of there is St. Philip's Anglican Church with its small cemetery. Further south still is Riverside Cemetery in the Humber Heights community, which I have argued strongly (nine months ago) is historically part of the greater Weston community. Could either of these two cemeteries have commingled with an earlier native burial site?

I don't know. But if we take Cruikshank and Nason at their word, there was an aboriginal burial place in this area and one doesn't have a burial place without an accompanying settlement. So, what evidence is there for a native settlement in Humber Heights? The Royal Ontario Museum has a number of native artifacts "from the Weston area". It is likely phrased that way because they weren't actually found within Weston, as the town came to be incorporated strictly on the east side of the river. Rather, they were from near there. There are records of an A.M. Kennedy, Esq., of Weston, having gifted a large number of native artifacts to the museum in 1914 and at least one of these came from a site "north of Weston" so I can't assume that any of his artifacts might have come from Humber Heights. In fact, I think it likely that all of Kennedy's finds might have come from a known Iroquois settlement north of Weston.

There does exist a record of a Weston archaeological site which was supposed to be a native "village", as "confirmed through surface sample and comments of local inhabitants" (Victor Konrad, 1972). Strangely, Konrad placed this site at the Humber Place apartments, in Weston, not in Humber Heights across the river. The Humber Place apartments were built in 1969, so Konrad was unable to carry out his own survey. If that was the case, how did he get a surface sample? He didn't. Apparently Victor Konrad learned of the site from a "Father Meighan" (from Upper Canada College). Father Meighan's name is attached to a number of Toronto area archaeological sites surveyed in 1950. It's possible that Humber Heights was one of those and that he is the one that did the surface sampling and talking to the locals. Konrad simply passed on that information. There is more. Konrad gave the site's location:

Cultural Matrix: 3-6 acres
Latitude: 434150
Longitude: 793140
Easting: 1930
Northing: 3925

Clearly Konrad did not come up with these numbers on his own as they place the village in Humber Heights. He must have transcribed them from Father Meighan's report! But if Meighan noted the location in Humber Heights, how (and why) did Konrad end up at the Humber Place apartments? It's a bit of a headscratcher. One possibility is that Meighan referred to the locale as "Weston" and a confused Konrad, aware only that the town of Weston had in 1967 been absorbed into the Borough of York, simply assumed (without referring to the site coordinates) that Meighan had meant the site to be east of the river and thus described it as so being there ("east of present arena").

There is a way to confirm or refute this hypothesis: Find Father Meighan's report on the site! Unfortunately this may not be possible. Apart from Father Meighan's name being associated with some Toronto archaeological surveys, I have been unable so far to find him in any other context. I haven't even been able to figure out his first name. And as this man has seemingly vanished from the face of history, so too has his report.

Nowadays, most folk peripherally acquainted with the matter will attest to the aboriginal "village" archaeological site as being at the Humber Place apartments and cite the nearby (less than 200 meters away on the other side of the Toronto Community Housing high-rise on Bellevue Crescent) "Weston bones" archaeological site as a native burial ground for it.

I must remind these people that the fourteen-or-so skeletons dug up there in 1911 constitute a site of unknown date and culture. The Toronto Historical Association suggests that they "may be French, since Indians rarely buried their dead in high-traffic areas" (Weston Road, the Carrying-Place trail, is 60 meters away). They could just as easily be English "who fought in the rebellion of 1837 or the war of 1812", as one Westonian suggested in a newspaper report at the time. We do know that Dr. Rowland B. Orr (Superintendent of the Provincial Museum), who visited that site more than three days after the bones had already been dug up with many of the skulls — by then — pilfered, almost certainly mischaracterized the skeletons to be "Indians" based on a supposed "arrangement" of the bones that is directly contradicted by eyewitness testimony.

Significantly perhaps, in the 1911 newspaper articles none of the Westonian gossip was to the effect of there having once been an Indian village close by!

Weston's archaeological sites

There are two archaeological sites in Weston and this morning I visited the Toronto Reference Library to see if I could document where exactly they might be. The book was Victor Konrad's 1973 "The archaeological resources of the Toronto planning area". I was more than a little disappointed with what little detail I found.

The two open circles represent Weston's two sites in the then- Borough of York, the upper-left one called "Scarlett Rd." (AkGv-5) and the lower-right one called "Bellevue Cr." (AkGv-6). I dealt with the Bellevue Crescent "Weston bones" site here, so we know the exact location of it. The circle in Konrad's map has it too far to the right by some 250 meters. I wonder if the Scarlett Rd. circle is similarly displaced. At the very least that would bring it closer to Scarlett Road which lies on the other side of the Humber River (the thick wavy line to the left of the circles).

Clearly the circles were not meant to be an accurate placement of the archaeological sites. Konrad lists the Scarlett Rd. site in the Borough of York (a 1967 municipality that absorbed the former town of Weston) but the actual Scarlett Road is not in that borough. Rather, it is in the adjacent municipality of Etobicoke. This discrepancy should be resolvable by a reference that notes the exact location of AkGv-5 but I can't seem to find an original source. It's not just the archaeological sites that have disappeared. Even the information about them is slowly becoming lost!