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DRAFT. This edition: January 28, 1998; First edition: September 19, 1997.

Equidistant Letter Sequences in Tolstoy's "War and Peace"

In [WRR1], Witztum, Rips and Rosenberg found a surprising correlation between famous rabbis and their dates of birth and death, as they appear as equidistant letter sequences in the Book of Genesis. We make a smaller or equal number of mistakes, and find the same phenomenon in Tolstoy's eternal creation "War and Peace".
Dror Bar-Natan
Institute of Mathematics
The Hebrew University
Giv'at-Ram, Jerusalem 91904
Brendan McKay
Department of Computer Science
Australian National University
Canberra, ACT, 0200

1. Introduction
    1.1. Acknowledgement
2. Their choices, our choices
    2.1. Modifications to specific appellations
    2.2. Modifications to the list of personalities
    2.3. Our list of appellations
3. The Results
4. Bibliography

1. Introduction

Our inspiration comes fully from reading the paper [WRR1] by Witztum, Rips and Rosenberg (WRR). The outline of the story in [WRR1] is as follows:

The purpose of this note is to show that WRR-Havlin still had some choice in applying their "rigid" procedures - enough choice to generate comparable significance levels in War and Peace. We do this by purposefully constructing our own list of appellations, staying within the WRR-stated rules or breaking them by about as much as they did. If one can find a list of appellations that works well on War and Peace (and pretty badly on Genesis), the only remaining reason to believe WRR-Havlin is personal trust in the cleanliness of the procedures they used to determine the list of appellations (and the other aspects of their experiment). While many beliefs are founded on trust bestowed on a very small group of prophets or apostles, such beliefs are often false, contradictory or absurd, and are definitely outside what one would normally call "science".

1.1. Acknowledgement

We wish to thank Maya Bar-Hillel, Menachem Cohen, Alec Gindis, Gil Kalai, Elchanan Reiner, Shlomo Sternberg, and the others who helped, for their suggestions and support.

2. Their choices, our choices

In [WRR1], WRR write:

The list of appellations for each personality was provided by Professor S.Z. Havlin, of the department of bibliography and librarianship at Bar Ilan University, on the basis of a computer search of the "Responsa" database at that university.
Contrary to what is suggested by the above quote, many of the appellations WRR do not even appear in the Bar-Ilan Responsa database [Re]. Thus in addition to the Responsa database [Re] we will also refer to the Margalioth Encyclopedia [Marg], used by WRR to select the rabbis, and to the highly-regarded Encyclopedia Hebraica [Heb] used by WRR in several of their other investigations. We (like WRR) also use other sources as needed.

2.1. Modifications to specific appellations

  1. We've added the appellation R)B"D $NY for Rabbi II-1 (the first rabbi in WRR's second list of personalities), Rabbi Avraham, Av-Beit-Din of Narbonne. A variant of this appellation that includes the definite article, HR)B"D H$NY, is the header of Rabbi Avraham's entry in [Az] It separates Rabbi II-1 from Rabbi I-1 (the first rabbi in WRR's first list of personalities), HR)B"D, the Ra'avad of Posquieres.
  2. The word HML)K (the angel) is an adjective, and not a last name. It is used in relation to several rabbis and even more often, it is used in relation with "real" angels (the angel of death, the angel Gabriel, the angel who struggled with Jacob, etc.). By itself, the word HML)K does not refer to Rabbi II-3, Rabbi Avraham. It is inconsistent to use this as an appellation for Rabbi Avraham, while at the same time not using HXSYD (the Hasid, the pious) for Rabbi I-11, Rabbi Yehuda haHasid (the word HXSYD at least always refers to a human...). Thus we remove the appellation HML)K.
  3. Rabbi II-6, Rabbi Eliezer Ashkenazi, is sometime called after his book titled "M($Y H". The variant M($Y Y/H/W/H is never written or pronounced, so we have removed it from the WRR lists and instead inserted the widely used appellations M($Y H and B(L M($Y H.
  4. According to [WRR1], they use grammatical orthography, KTYB DQDWQY, for the spelling of Hebrew words. This rule cannot be applied to the last name of Rabbi II-7, Rabbi David Oppenheim, whose origin is in the Yiddish or German language and not in Hebrew. Yiddish words are spelled in Hebrew letters, and, WRR say, "there is no need to transliterate" them. Hence we replace their )WPNHYM (Oppenhim) with the commonly used form, )WPNHYYM, which is used in their source encyclopedia [Marg]. In Responsa, )WPNHYM appears once, while )WPNHYYM appears over 50 times, including a number of times as Rabbi David Oppenheim's own signature! See also [Az] and [Ab, page 40, 1st col., bottom].
  5. Widely used acronyms sometimes acquire the status of a word, and are used as if they were a word. A good example is the acronym/word AIDS. When reading it, we do not expand it to "Acquired Immune Deficiency Syndrome". Following common practice, WRR use such "pronounced" acronyms as appellations. Rabbi I-29, the Rambam, is best known as HRMB"M, and not by his full name, and it is reasonably included as an appellation in their first sample. But they are inconsistent about the use of acronyms that did not attain the status of a word. They have used HR)B"Y, B(L HLQ"+, HRY(B"C, )"X H("R and a few other such acronyms, but they have left out acronyms such as HRB Z"), MHRY"M, MHRX"), etc. We have done the same, removing one of their acronyms and adding two new ones: we have removed HRY(B"C (Rabbi II-24, the Yaabez), and added HMHRX") and MHRX") (Rabbi II-10, Rabbi Haim Abulafia). The last acronym appears very often in the Bar-Ilan Responsa Database [Re], it is used in [Az], and its omission in [WRR1] is especially questionable.

    Rabbi Haim Abulafia and 6 of Nisan
    The convergence of HMHRX") and his date of death, W NYSN, in War and Peace.

  6. In [Hav], Prof. Havlin described the methodology he employed in producing the list of appellations. In his report he acknowledged a few omissions he made in the original list. One of those omissions is the appellation HRB XBY"B for Rabbi II-11, Rabbi Haim Benvenist(e). Thus we have added HRB XBY"B to our list, with and without the definite article H. We have also added the widely used HRB HXBY"B.
  7. The appellations BNBN$T and BNBN$TY are both used for Rabbi II-11, Rabbi Haim Benvenist(e). WRR chose BNBN$T, we choose BNBN$TY. (Incidentally, BNBN$TY appears in Responsa more than BNBN$T, in roughly a 3:1 ratio).
  8. The last name of Rabbi II-12, Rabbi Haim Capusi, can be spelled either KPWSY or K)PWSY (In Responsa, they appear in an 8:3 ratio, which indicates quite clearly that both spellings are valid). WRR state explicitly in [WRR1] that in such cases they use both spellings. But in their list they took only the first form. For our list, we make the opposite mistake and take only the second form.
  9. The two forms B(L NS and B(L HNS are most often associated with Rabbi Meir, whose grave is near Tiberias, rather than with Rabbi II-12, Rabbi Haim Capusi. The vast majority of references to B(L HNS in Responsa refer to Rabbi Meir and not to Rabbi Haim Capusi, and we found no references to Rabbi Haim Capusi as "B(L NS". (Responsa does refer to this Rabbi often; see the previous point). We note that on several occasions Havlin ruled out an appellation for a certain personality because it was more closely associated with some other personality (see [Hav]). Hence we remove the appellations B(L NS and B(L HNS.
  10. Rabbi II-15 is Rabbi Yehuda Hasid Segal. WRR omitted the appellation YHWDH SG"L, his first name + his last name (notice that according to [WRR2], WRR always take appellations of this form, when it is available). So we add YHWDH SG"L to our list. See also [Marg].
  11. We add the appellation HR"Y +R)NY to Rabbi II-19, The Maharit, along with the variations R"Y +RNY, HR"Y +RNY, and R"Y +R)NY. This puts the Maharit in a similar status with the Yaabez.

    The Maharit and 14 of Tamuz The convergence of +R)NY
    and his date of death,
    Y"D TMWZ, in War and Peace.

  12. We've added the appellation MYMWNGYL for Rabbi II-21, Rabbi Yaakov Beirav. It is his last name by his own testimony, see [Be, S' )']. See also the article [Gr], where the same source ([Be, S' )']) was used to determine Rabbi Beirav's last name. [Gr] says the name is MY)NGYL. Given the explicit "MYMWNGYL" in [Be, S' )'], the "MY)NGYL" in [Gr] must be a typographical error.

    Rabbi Yaakov Beirav and 30 of Nisan
    The convergence of HRYB"R and his date of death, L BNYSN, in War and Peace.

  13. The last name of Rabbi II-22, Rabbi Israel Yaakov Hagiz, can be spelled either XGYZ (as in [Heb] and as in the biographical section of [Re]) or X)GYZ (as in [Marg]). But WRR use only X)GYZ, contrary to their explicit convention that where ) is used as a "mater lectionis", they take both forms. Thus they fail to use the appellations MHR"Y XGYZ and R"Y XGYZ (both appear in Responsa). We allow ourselves to make the opposite mistake, taking the appellations MHR"Y XGYZ and R"Y XGYZ and omitting X)GYZ. We note that even though the WRR computations are restricted to appellations totaling 5-8 letters, we can tell which `short' appellations (such as XGYZ, (MDN, or MWLN) they consider as valid either by checking whether they have used longer appellations that contain the shorter ones as substrings or by reading their "blue preprint" [WRR2], in which the short forms are also listed.
  14. For Rabbi I-31, The Shach, WRR use both the appellation $BTY KHN and the appellation $BTY HKHN. We follow suit, and for Rabbi II-23, the Maharil and Rabbi II-25, Rabbi Yitshak Horowitz, we use both LWY and HLWY. Thus for Rabbi II-23 we add the appellations Y(QB LWY and MHR"Y LWY on top of the existing Y(QB HLWY and MHR"Y HLWY, and for Rabbi II-25 we add YCXQ LWY on top of the existing YCXQ HLWY.
  15. In the case of Rabbi II-24, the Yaabez, we do not use appellations based around the spelling (MDN for two reasons. They appear less often, and we wish to follow the precedent set by WRR when they did not use the form MWLN with Rabbi II-23, the Maharil. (The appellations Y(QB MWLN, MHR"Y MWLYN, etc., appear often in Responsa, more often than forms with (MDN, and were omitted in [WRR1]. See also [WRR2]).

    We note that there may be a case for removing the name (MDYN altogether, for it is just the name of a town were the Yaabez was briefly a Rabbi, and not his last name. The Yaabez himself wrote in [Ya],

    )NY $WQD (L TQNT'...
    In free translation to English, this reads:
    ... The busy Yaakov Israel known as Yaabez good omen was never called Yaakov Emdyn ((MDYN) (as has done the honourable in the address on this letter as is the deplorable habit in the tongues of the people). It is known I am not from the people of Emdyn, was not born there, do not expect to see it, but am looking after its welfare and benefit as in the former times when I was under it's load, [and] so also am I now diligent for it's remedy. ...
    Little did his plea help, and the Yaabez has several common appellations which are variants of the word (MDYN. We keep them in our list.
  16. WRR are inconsistent about the use of the definite article, H. For example, they use HR"Y (MDN and HR"Y (MDYN for Rabbi II-24, but omit R"Y (MDN and R"Y (MDYN. (The latter two forms appear in Responsa more often than the former two!) We fix this mistake and allow ourselves to make a parallel mistake, and omit the appellation HMHR$"$ for Rabbi II-31, Rabbi Shalom Sharabi, while keeping the appellation MHR$"$.
  17. The last name of Rabbi II-25, Rabbi Yitshak HaLevi Horowitz, is spelled HWRWBYC by both [Marg] and [Heb]. We thus replace HWRWWYC by HWRWBYC.
  18. The Krochmal Story: We suspected that there's something wrong with the name QRWKML for Rabbi II-26, Rabbi Menachem Mendel Krochmal, author of the responsa book CMX CDQ and of the biblical commentary PY CDYQ. So we started searching. The word QRWKML does not appear in the Bar-Ilan Responsa database [Re]. The only variation of QRWKML that does appear there is QR)KM)L, which appears only twice. But in both appearances it is the Yiddish word for starch, and not a Jewish surname. And it is not because [Re] doesn't care about Rabbi Menachem Mendel; the Rabbi appears in [Re] many times, but only under his other common designations. In [Marg], QRWKML is in the header of Rabbi Menachem Mendel's entry, but no explanation for the origin of the name is given. In [Heb] Menachem Mendel doesn't even have an entry, though the index mentions him twice. Looking inside the text, we got no further clues. His books are signed MNXM M(NDL and MNXM M(NDYL, and no QRWKML is mentioned in them in any form. An eulogy for him [Shm] mentions only his first name(s), MNXM M(NDYL, and the titles of his books. So where did "QRWKML" come from? [Az] was of no help. In the 19th century bibliography [B-Y] one of Krochmal's books is listed under QR)KM(L. In a 19th century biography [Du] of David Oppenheim, Krochmal is mentioned in passing, and his name is given as QR)KM)L. We are almost certain we also saw QRWKM)L and QRWKM(L, but after a long day in the Israeli National Library in Giv'at-Ram, the stairs leading to the photocopy machines seem very steep and the pencil becomes really heavy, so we don't have references for these two forms. The 19th century Rabbi Nahman Krochmal spells his name QRWKM)L. All of that taken together indicates clearly that Krochmal was not spelled "QRWKML" in the 19th century, but it doesn't help us find how Krochmal was spelled when Rabbi Menachem Mendel lived, in the first part of the 17th century.

    At this point we got the advice of two wise men. One suggested that we look at [Hei], a book on the Jewish laws in the state of Moravia, where Rabbi Menachem Mendel was the state's Rabbi. From the other wise man we learned to check the citations in the footnotes. One footnote, on page 111 of [Hei], he checked himself. It lead to an article [Marx], that contain a letter written by the son of a nephew of Rabbi Menachem Mendel in the late 17th century, only a few dozen years after Rabbi Menachem Mendel died in 1661. In that letter Rabbi Menachem Mendel's surname is given as QR)XM)L. The following day (and a continent away), we checked the footnote on page 102. It lead us to two articles, [Har] and [Ka], devoted to our Rabbi and his descendents. Both articles use the spelling QR)KM)L extensively and not just in passing, and the latter one even explains where the name comes from! It relates the name QR)KM)L to a certain earlier Dayan, Rabbi Jonah Krochmals in the city of Cracow, where Rabbi Menachem Mendel was born. A transcript of the tombstone of Rabbi Jonah Krochmals is given in [Zo, page 180] and Krochmals is spelled QR)KM)L$ there. Thus there is no doubt that the original spelling of the name Krochmal is QR)KM)L and we remove the appellation QRWKML from our list, putting QR)KM)L instead.

    Note that here we corrected new to old, while in the case of Rabbi II-25, Rabbi Yitshak HaLevi Horowitz we corrected old to new. Whatever inconsistencies WRR have we are allowed to have too.

    We also note that once it is clear that QR)KM)L is an acceptable spelling for Krochmal, the usage of this spelling is mandatory according to the WRR rules, which state explicitly that Yiddish names are spelled as in the original Yiddish. Recall that QR)KM)L is a Yiddish word.

  19. The last name of Rabbi II-27, Rabbi Moshe Zacut, is ZKWT, and not ZKWT) or ZKWTW. See his own signatures in his book QWL HRM"Z, see his biography [Ap], and see [Marg], [Heb], and [Az]. Hence we remove the appellations ZKWT), ZKWTW, M$H ZKWT), and M$H ZKWTW.
  20. Often great rabbis are called after their books. Thus we add the appellation HWN ($YR for Rabbi II-30, Rabbi Immanuel Hai Ricchi. We comment here that the appellation HWN ($YR appears often in [Re], while the appellation Y$R LBB does not appear there at all, not even in its correct form, YW$R LBB (the phrase YW$R LBB does appear several times, sometimes even in reference to Rabbi Ricchi's book. But it is never used as an appellation).
  21. The story of the appellation )"X H("R of Rabbi II-30, Rabbi Immanuel Hai Ricchi, is particularly telling. First, we couldn't find it anywhere, and nobody we asked could tell us what it meant. When we inquired with Doron Witztum, he said Rabbi Ricchi used it as his signature in some of his books, and that it expands to ")NY XY, HC(YR (MNW)L RYQY" (I'm alive, the young Immanuel Ricchi). We think the inclusion of such an acronym is extremely silly. It is a signature; not an appellation. Nobody should refer to Rabbi Ricchi by this name other than himself, not even the author of the hidden codes in the Book of Genesis (see a parrallel in [Ab, page 40, 2nd column, top]). Anyway, it seems that nobody does refer to Rabbi Ricchi by this name, for we could find no such references and we could find no one who even knows what it means! In particular, it is not "pronounced", NHGH.

    The story doesn't end there. When we tried to find )"X H("R as a signature in Rabbi Ricchi's books, we failed. What we did find was a different permutation of these letters, )H("XR, which appears with its expansion, )NY HC(YR (MNW)L XY RYQY (me the young, Immanuel Hai Ricchi). We note that in Hebrew the latter expansion makes much more sense than the former, and that the dictionary of acronyms [AY] lists the latter acronym but not the former. Including ")H("XR" as an appellation is senseless for the same reasons as above, but it still makes more sense than including ")"X H("R". So we deleted ")"X H("R" and inserted ")H("XR".

    After the first version of this note was widely circulated, we finally found a single reference to the variation )"X H("R, in [Shi]. That source refers to an acronym that appears as a signature on an addendum, only available in Zurich, to Rabbi Ricchi's book "M($H XW$B". Neither the original edition of that book, nor any of his other books that we checked, contain that acronym; the original edition of "M($H XW$B", like many of Rabbi Ricchi's other books, contains )H("XR. We don't have the resources to check the manuscript in Zurich, so at the moment we don't know if the acronym )"X H("R really appears there or if [Shi] copied it with the X misplaced. However, we can see two things clearly: if the acronym )"X H("R exists at all it is a signature and not an appellation, and the WRR claim to have admitted only widely used pronounced acronyms is false.

  22. The sources [Heb], [Marg] , [Az], and [Re] never refer to II-31, Rabbi Shalom Sharabi, as $R $LWM, and we found at least 7 other rabbis that carry the name $R $LWM (and that are referred to by this name extensively). Some of these 7 are much earlier than Rabbi II-31. Thus the appellation $R $LWM certainly does not identify $R $LWM and hence we remove it. WRR do the same in similar situations; see [Hav].
  23. The appellation MZRXY for Rabbi II-31 is identified with a Rabbi from the Rishonim. We remove it. Notice that MZRXY is not the last name of Rabbi Sharabi, and is related to him just like the name )$KNZY relates to Rabbi I-6, Rabbi Gershon Ashkenazi and to Rabbi II-6, Rabbi Eliezer Ashkenazi. In both those cases WRR did not use the appellation )$KNZY.
  24. The last name of Rabbi II-32, Rabbi Shelomo Chelma, can be spelled either as XLM) or as X(LM). See the header to his entry in WRR's source encyclopedia, [Marg], and see his biography [Br] (one can say that Rabbi Chelma's last name is X(LM), and that the letter ( is "mater lectionis"). Thus we wish to add the appellations XLM), X(LM), $LMH XLM), and $LMH X(LM). In practice we only add X(LM) and $LMH XLM), for the other two appellations do not fit within 5-8 letters.

    Rabbi Chelma and 21 of Tamuz
    The convergence of X(LM) and his date of death, K") BTMWZ, in War and Peace.

2.2. Modifications to the list of personalities

We make the following additional changes to the list produced by WRR:

2.3. Our list of appellations

The resulting new list of appellations is given in the table below:

The old and new lists, a comparative table
# Original entry we remove we add new entry
1 H)$KWL, HR)B"D, HR)B"Y,
  R)B"D $NY H)$KWL, HR)B"D, HR)B"Y,
  --- completely removed ---
6 M($Y H$M, M($Y Y/H/W/H M($Y Y/H/W/H B(L M($Y H', M($Y
B(L M($Y H', M($Y H', M($Y
8DWD HNGYD, RBY DWD  --- completely removed ---
14 XWT Y)YR     XWT Y)YR
19 HMHRY"+, HMHRYM"+, +R)NY,
  HR"Y +R)NY, HR"Y
+RNY, R"Y +R)NY,
+R)NY, HR"Y +RNY, +R)NY,
20PRY MGDYM, RBY YWSP, T)WMYM  --- completely removed ---
24 HY(B"C, HRY(B"C, HR"Y (MDN,
30 )"X H("R, Y$R LBB )"X H("R )H("XR, HWN ($YR )H("XR, HWN ($YR, Y$R LBB
31 HMHR$"$, MHR$"$, MZRXY,
  MHR$"$, RBY $LWM, $R(BY
33--- new Rabbi ---  )YZN$+)+, )YZN$++, )YYZN$++,

We have shown our list of appellations to Professor Menachem Cohen, of the Bible Department at Bar-Ilan University, Israel. In reference to our list and to the original WRR-Havlin list he wrote in [Co2]:

... I see no essential difference between the two lists for the purpose of using them for skip experiments in any text.

3. The Results

The table below contains the permutation test ranks obtained by running our list against the same list of dates as in [WRR1] (with the addition of K"Z SYWN, BK"Z SYWN, and K"Z BSYWN for Rabbi Meir Eisenstadt), on an initial segment of War and Peace which is of the same length as Genesis, using the same four computation methods (P1-P4) used in [WRR1]. For comparison, we also include the scores obtained by running our list on Genesis and the scores reported by WRR in [WRR1] for their list on Genesis.

Permutation test ranks out of 106

P1 P2 P3 P4
our list on War and Peace 620 19 31 2
our list on Genesis 201,278 13,099 294,296 9,524
the WRR list on Genesis 453 5 570 4

Comparing the first and the last row, we see that our list does as well on War and Peace as the WRR list does on Genesis. Tolstoy would have enjoyed knowing that. Some of the numbers in the middle row are "middle of the way", smallish but not very small. They are "smallish" because our list is still highly correlated with the original WRR list, on which it was based. The fact that they are not very small needs to be explained by WRR, not by us. Why is it that an equally valid list of appellations (our list) does so much worse than their list on Genesis?

Comment: The computations of the significance levels for our list was carried out using a program WRR gave us, els2.c. We modified it to work under Unix and re-wrote the permutation test part, but made no modifications to the main part of the code. The text we used was also given to us by WRR, and consists of the first 78,064 letters (the length of Genesis) of a Hebrew translation of Tolstoy's War and Peace.

4. Bibliography

Yehiel Abramson, YXY)L )BRMSWN, The battle over the code, HQRB (L HCWPN, Mishpaha, M$PXH, 307 (July 3, 1997) 10-13,40. (This article is a collection of lies and distortions, but we agree with its author that using signatures is silly and we like its use of the spelling ")WPNHYYM").
Abe Apelbaum )B) )PPLBWYM, biography of Rabbi Moshe Zacut titled M$H ZKWT, Snunit, Lvov 1925.
Shmuel Ashkenazi, $MW)L )$KNZY, and Dov Yarden, DB YRDN, a dictionary of Hebrew acronyms titled )WCR R)$Y TBWT.
Azulai, Rabbi Haim Yosef David, Shem haGdolim haShalem, $M HGDWLYM H$LM, Jerusalem 1979.
Rabbi Yaakov Beirav, Beirav Responsa, $)LWT WT$WBWT LY(QB BY RB, Tif'ereth haTorah, Jerusalem 1988.
Yitshak Isaac Ben-Yaakov, YCXQ )YYZYQ BN-Y(QB, a bibliography of Hebrew books titled )WCR HSPRYM, 1880.
Avraham Brik, )BRHM BRYQ, biography of Rabbi Shelomo Chelma titled R' $LMH X(LM), B(L "MRKBWT HM$NH".
Menachem Cohen, A letter to Dror Bar-Natan dated September 2, 1997. Available at http://cs.anu.edu.au/~bdm/dilugim/Cohen.html.
Menachem Cohen, A letter to Dror Bar-Natan dated October 27, 1997. Available at http://cs.anu.edu.au/~bdm/dilugim/WNP/CohenLetter2.html.
Yaakov Kapel Duschinski, Y(QB Q)PL DW$YNSQY), biography of Rabbi David Oppenheim titled TWLDWT HG)WN R' DWD )WPNHYYM(R ZC"L.
Eliezer HaLevi Grinhut, )LY(ZR HLWY GRYNHW+, biography of of Rabbi Yaakov Beirav titled LTWLDWT HG)WN RB Y(QB BY RB, in HCWPH M)RC HGR, year 2, Budapest HTR("B.
Shmuel Abe Haradsky, $MW)L )B) H)R)D(CQY, biography of Rabbi Menachem Mendel Krochmal titled RBY MNXM MNDL QR)KM)L, Hagoren, HGRN, 2 (1900) 32.
Shlomo Z. Havlin, a brief dated October 30, 1996. English translation (by WRR) available at http://www.torahcodes.co.il/torasages.pdf.
Hebraica Encyclopedia, H)NCYQLWPDYH H(BRYT.
Israel Heilperin, Y$R)L HYYLPRYN, Regulations of the state of Moravia, TQNWT MDYNT M(HRYN.
David Kauffman, DWD QWYPM)NN, biography of Rabbi Menachem Mendel Krochmal and his family titled LDWD CMX, T$WBT B(L CMX CDQ WQCWR TWLDWT M$PXTW, Hagoren, HGRN, 2 (1900) 38.
M. Margalioth, Encyclopedia of Great Man in Israel; a Bibliographical Dictionary of Jewish Sages and Scholars from the 9th to the End of the 18th Century, vols. 1-4, Joshua Chachik, Tel Aviv.
Alexander Marx, A seventeenth-century autobiography, Jewish Quarterly Review 8 (1917-8) 269.
The Bar-Ilan Responsa CD-ROM, version 4, Bar-Ilan University, Israel.
Rabbi A. Shisha HaLevi, Rabbi Immanuel Hai Ricchi and his book DQDWQY (NYWT-HWN ($YR, Jerusalem, 1972.
Shmarya Shmaril, $MRYH $M(RYL, an eulogy for Rabbi Menachem Mendel Krochmal titled QWN+RS ZKR (WLM, 1885.
D. Witztum, E. Rips and Y. Rosenberg, Equidistant Letter Sequences in the Book of Genesis, Statistical Science 9-3 (1994) 429-438.
D. Witztum, E. Rips and Y. Rosenberg, Equidistant Letter Sequences in the Book of Genesis, "The Blue Preprint", an older version of [WRR1] (1987).
The Yaabez, The Yaabez Responsa, $W"T $)YLT Y(B"C, part B sub 24.
Yekhiel Matityahu Zonz, YXY)L MTTYHW CWNC, history of the rabbis of Cracow titled (YR HCDQ.

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© Copyright (1997) Dror Bar-Natan, drorbn@math.huji.ac.il and Brendan McKay, bdm@cs.anu.edu.au.