The following is an explanation of our extension of Brocard's Problem or Brocard's Enigma. Regarding the following 14 equations
0! + 02 = 12
1! + 02 = 12
4! + 12 = 52
5! + 12 = 112
6! + 32 = 272 = (3*9)2
7! + 12 = 712
8! + (3*3)2 = 2012 = (3*67)2
9! + (9*3)2 = 6032 = (9*67)2
10! + 152 = 19052 = (15*127)2
11! + 182 = 63182 = (18*351)2
13! + 2882 = 789122 = (288*274)2
14! + 4202 = 2952602 = (420*703)2
15! + (16*29)2 = 11435362 = (16*71471)2
16! + (64*29)2 = 45741442 = (64*71471)2
I postulated almost seven years ago that
D(n) = (ceiling(sqrt(n!)))2 - n!
cannot be a square number if n is greater than 16. What does that mean?
For example, take n=6. sqrt(6!) = sqrt(720) = 26.83282. The ceiling of a number is the smallest integer number above it, or equal to it. In our case, ceiling(sqrt(6!)) = 27. The square of 27 is 729, which is 9 more that 6!. So, D(6) = 9, and it is a square, of 3. Brocard's observation, in our language, is that D(n) = 1 for n = 4, for n = 5, and for n = 7, and after that, seemingly, D(n) ≠ 1 forever.
It happens that D(n) is a square for 14 out of the first 17 integer numbers, detailed above. And after that it is a square no more, according to the postulate, not even once. If the above are the only 14 cases of n for which D(n) is a square, D(n) is equal to 1 exactly three times. Thus, if the postulate is true, it solves the problem of Henri Brocard, and proves the conjecture of Paul Erdős.
Regarding proofs, we know from Erdős, among others, that D(n) ≠ 0, for n > 1. As far as I know, for the remainder, the problem is open. For some more info, see this.
Monday, March 28, 2016
Tuesday, March 22, 2016
נ"ר תמי"ד: זַיִ"ת זָ"ךְ כָּ"תִ י"ת
נ"ר תמי"ד: זַיִ"ת זָ"ךְ כָּ"תִ י"ת
בְּרֵאשִׁית, בָּרָא אֱלֹהִים, אֵת הַשָּׁמַיִם, וְאֵת הָאָרֶץ. וְהָאָרֶץ, הָיְתָה תֹהוּ וָבֹהוּ, וְחֹשֶׁךְ, עַל-פְּנֵי תְהוֹם; וְרוּחַ אֱלֹהִים, מְרַחֶפֶת עַל-פְּנֵי הַמָּיִם. וַיֹּאמֶר אֱלֹהִים, יְהִי אוֹר; וַיְהִי-אוֹר. וַיַּרְא אֱלֹהִים אֶת-הָאוֹר, כִּי-טוֹב; וַיַּבְדֵּל אֱלֹהִים, בֵּין הָאוֹר וּבֵין הַחֹשֶׁךְ. וַיִּקְרָא אֱלֹהִים לָאוֹר יוֹם, וְלַחֹשֶׁךְ קָרָא לָיְלָה; וַיְהִי-עֶרֶב וַיְהִי-בֹקֶר, יוֹם אֶחָד. וַיְכֻלּוּ הַשָּׁמַיִם וְהָאָרֶץ, וְכָל-צְבָאָם, בְּיוֹם, עֲשׂוֹת יְהוָה אֱלֹהִים--אֶרֶץ וְשָׁמָיִם:
הָיָה דְבַר-יְהוָה אֶל-אַבְרָהָם, בַּמַּחֲזֶה, לֵאמֹר: אַל-תִּירָא אַבְרָהָם, אָנֹכִי מָגֵן לָךְ--שְׂכָרְךָ, הַרְבֵּה מְאֹד. וְהֶאֱמִן, בַּיהוָה; וַיַּחְשְׁבֶהָ לּוֹ, צְדָקָה. וַיֹּאמֶר, אֵלָיו: אֲנִי יְהוָה, אֲשֶׁר הוֹצֵאתִיךָ מֵחָרָן--לָתֶת לְךָ אֶת-הָאָרֶץ הַזֹּאת, לְרִשְׁתָּהּ. וַיֹּאמַר: אֲדֹנָי יְהוִה, בַּמָּה אֵדַע כִּי אִירָשֶׁנָּה. וַיֹּאמֶר אֵלָיו, קְחָה לִי עֶגְלָה מְשֻׁלֶּשֶׁת, וְעֵז מְשֻׁלֶּשֶׁת, וְאַיִל מְשֻׁלָּשׁ; וְתֹר, וְגוֹזָל. וַיִּקַּח-לוֹ אֶת-כָּל-אֵלֶּה, וַיְבַתֵּר אֹתָם בַּתָּוֶךְ, וַיִּתֵּן אִישׁ-בִּתְרוֹ, לִקְרַאת רֵעֵהוּ; וְאֶת-הַצִּפֹּר, לֹא בָתָר. וַיֵּרֶד הָעַיִט, עַל-הַפְּגָרִים; וַיַּשֵּׁב אֹתָם, אַבְרָהָם. וַיְהִי הַשֶּׁמֶשׁ לָבוֹא, וְתַרְדֵּמָה נָפְלָה עַל-אַבְרָהָם; וְהִנֵּה אֵימָה חֲשֵׁכָה גְדֹלָה, נֹפֶלֶת עָלָיו. וַיֹּאמֶר לְאַבְרָהָם, יָדֹעַ תֵּדַע כִּי-גֵר יִהְיֶה זַרְעֲךָ בְּאֶרֶץ לֹא לָהֶם, וַעֲבָדוּם, וְעִנּוּ אֹתָם--אַרְבַּע מֵאוֹת, שָׁנָה. וְגַם אֶת-הַגּוֹי אֲשֶׁר יַעֲבֹדוּ, דָּן אָנֹכִי; וְאַחֲרֵי-כֵן יֵצְאוּ, בִּרְכֻשׁ גָּדוֹל. וְאַתָּה תָּבוֹא אֶל-אֲבֹתֶיךָ, בְּשָׁלוֹם: תִּקָּבֵר, בְּשֵׂיבָה טוֹבָה. וְדוֹר רְבִיעִי, יָשׁוּבוּ הֵנָּה: כִּי לֹא-שָׁלֵם עֲוֹן הָאֱמֹרִי, עַד-הֵנָּה.
הַשָּׂדֶה אֲשֶׁר-קָנָה אַבְרָהָם, מֵאֵת בְּנֵי-חֵת--שָׁמָּה קֻבַּר אַבְרָהָם, וְשָׂרָה אִשְׁתּוֹ. וַיִּהְיוּ חַיֵּי שָׂרָה, מֵאָה שָׁנָה וְעֶשְׂרִים שָׁנָה וְשֶׁבַע שָׁנִים--שְׁנֵי, חַיֵּי שָׂרָה. וְאֵלֶּה, יְמֵי שְׁנֵי-חַיֵּי אַבְרָהָם--אֲשֶׁר-חָי מְאַת שָׁנָה וְשִׁבְעִים שָׁנָה, וְחָמֵשׁ שָׁנִים, וַיִּגְוַע. וַיִּקְבְּרוּ אֹתוֹ יִצְחָק וְיִשְׁמָעֵאל, בָּנָיו, אֶל-מְעָרַת, הַמַּכְפֵּלָה: אֶל-שְׂדֵה עֶפְרֹן בֶּן-צֹחַר, הַחִתִּי, אֲשֶׁר, עַל-פְּנֵי מַמְרֵא. וַיָּקָם הַשָּׂדֶה וְהַמְּעָרָה אֲשֶׁר-בּוֹ, לְאַבְרָהָם--לַאֲחֻזַּת-קָבֶר: מֵאֵת, בְּנֵי-חֵת.
וַיְהִי אַחֲרֵי מוֹת אַבְרָהָם וַיְבָרֶךְ אֱלֹהִים אֶת יִצְחָק בְּנוֹ וַיֵּשֶׁב יִצְחָק, עִם-בְּאֵר לַחַי רֹאִי. וַיִּשְׁכְּן יִשְׁמָעֵאל מֵחֲוִילָה עַד-שׁוּר, אֲשֶׁר עַל-פְּנֵי מִצְרַיִם, בֹּאֲכָה, אַשּׁוּרָה; עַל-פְּנֵי כָל-אֶחָיו ,נָפָל. וְאֵלֶּה, שְׁנֵי חַיֵּי יִשְׁמָעֵאל--מְאַת שָׁנָה וּשְׁלֹשִׁים שָׁנָה, וְשֶׁבַע שָׁנִים; וַיִּגְוַע. וַיֶּעְתַּר יִצְחָק לַיהוָה לְנֹכַח אִשְׁתּוֹ, כִּי עֲקָרָה הִוא; וַיֵּעָתֶר לוֹ יְהוָה, וַתַּהַר רִבְקָה אִשְׁתּוֹ. וַיִּתְרֹצְצוּ הַבָּנִים, בְּקִרְבָּהּ, וַתֹּאמֶר אִם-כֵּן, לָמָּה זֶּה אָנֹכִי; וַתֵּלֶךְ, לִדְרֹשׁ אֶת-יְהוָה. וַיֹּאמֶר יְהוָה לָהּ, שְׁנֵי גֹיִים בְּבִטְנֵךְ, וּשְׁנֵי לְאֻמִּים, מִמֵּעַיִךְ יִפָּרֵדוּ; וּלְאֹם מִלְאֹם יֶאֱמָץ, וְרַב יַעֲבֹד צָעִיר. וַיִּמְלְאוּ יָמֶיהָ, לָלֶדֶת; וְהִנֵּה תוֹמִם, בְּבִטְנָהּ. וַיֵּצֵא הָרִאשׁוֹן אַדְמוֹנִי, כֻּלּוֹ כְּאַדֶּרֶת שֵׂעָר; וַיִּקְרְאוּ שְׁמוֹ, עֵשָׂו. וְאַחֲרֵי-כֵן יָצָא אָחִיו, וְיָדוֹ אֹחֶזֶת בַּעֲקֵב עֵשָׂו, וַיִּקְרָא שְׁמוֹ, יַעֲקֹב; וְיִצְחָק בֶּן-שִׁשִּׁים שָׁנָה, בְּלֶדֶת אֹתָם. וַיִּהְיוּ, יְמֵי יִצְחָק--מְאַת שָׁנָה, וּשְׁמֹנִים שָׁנָה, וַיִּגְוַע. וַיִּקְבְּרוּ אֹתוֹ, עֵשָׂו וְיַעֲקֹב בָּנָיו.
וַיֶּאֱהַב יַעֲקֹב, אֶת-רָחֵל; וַיַּעֲבֹד יַעֲקֹב בְּרָחֵל, שֶׁבַע שָׁנִים; וַיִּהְיוּ בְעֵינָיו כְּיָמִים אֲחָדִים, בְּאַהֲבָתוֹ אֹתָהּ. וַיִּזְכֹּר אֱלֹהִים, אֶת-רָחֵל; וַיִּשְׁמַע אֵלֶיהָ אֱלֹהִים, וַיִּפְתַּח אֶת-רַחְמָהּ. וַתַּהַר, וַתֵּלֶד בֵּן; וַתֹּאמֶר, אָסַף אֱלֹהִים אֶת-חֶרְפָּתִי. וַתִּקְרָא אֶת-שְׁמוֹ יוֹסֵף. וַיָּבֵא יוֹסֵף אֶת-יַעֲקֹב אָבִיו, וַיַּעֲמִדֵהוּ לִפְנֵי פַרְעֹה; וַיְבָרֶךְ יַעֲקֹב, אֶת-פַּרְעֹה. וַיֹּאמֶר פַּרְעֹה, אֶל-יַעֲקֹב: כַּמָּה, יְמֵי שְׁנֵי חַיֶּיךָ. וַיֹּאמֶר יַעֲקֹב, אֶל-פַּרְעֹה, יְמֵי שְׁנֵי מְגוּרַי, שְׁלֹשִׁים וּמְאַת שָׁנָה: מְעַט וְרָעִים, הָיוּ יְמֵי שְׁנֵי חַיַּי, וְלֹא הִשִּׂיגוּ אֶת-יְמֵי שְׁנֵי חַיֵּי אֲבֹתַי, בִּימֵי מְגוּרֵיהֶם. וַיְבָרֶךְ יַעֲקֹב, אֶת-פַּרְעֹה; וַיֵּצֵא, מִלִּפְנֵי פַרְעֹה. וַיּוֹשֵׁב יוֹסֵף, אֶת-אָבִיו וְאֶת-אֶחָיו, וַיִּתֵּן לָהֶם אֲחֻזָּה בְּאֶרֶץ מִצְרַיִם, בְּמֵיטַב הָאָרֶץ בְּאֶרֶץ רַעְמְסֵס. וַיְכַלְכֵּל יוֹסֵף אֶת-אָבִיו וְאֶת-אֶחָיו, וְאֵת כָּל-בֵּית אָבִיו--לֶחֶם, לְפִי הַטָּף. ויִּפְרוּ וַיִּרְבּוּ מְאֹד. וַיְחִי יַעֲקֹב בְּאֶרֶץ מִצְרַיִם, שְׁבַע עֶשְׂרֵה שָׁנָה; וַיְהִי יְמֵי-יַעֲקֹב, שְׁנֵי חַיָּיו--שֶׁבַע שָׁנִים, וְאַרְבָּעִים וּמְאַת שָׁנָה, וַיִּגְוַע
בְּרֵאשִׁית, בָּרָא אֱלֹהִים, אֵת הַשָּׁמַיִם, וְאֵת הָאָרֶץ. וְהָאָרֶץ, הָיְתָה תֹהוּ וָבֹהוּ, וְחֹשֶׁךְ, עַל-פְּנֵי תְהוֹם; וְרוּחַ אֱלֹהִים, מְרַחֶפֶת עַל-פְּנֵי הַמָּיִם. וַיֹּאמֶר אֱלֹהִים, יְהִי אוֹר; וַיְהִי-אוֹר. וַיַּרְא אֱלֹהִים אֶת-הָאוֹר, כִּי-טוֹב; וַיַּבְדֵּל אֱלֹהִים, בֵּין הָאוֹר וּבֵין הַחֹשֶׁךְ. וַיִּקְרָא אֱלֹהִים לָאוֹר יוֹם, וְלַחֹשֶׁךְ קָרָא לָיְלָה; וַיְהִי-עֶרֶב וַיְהִי-בֹקֶר, יוֹם אֶחָד. וַיְכֻלּוּ הַשָּׁמַיִם וְהָאָרֶץ, וְכָל-צְבָאָם, בְּיוֹם, עֲשׂוֹת יְהוָה אֱלֹהִים--אֶרֶץ וְשָׁמָיִם:
הָיָה דְבַר-יְהוָה אֶל-אַבְרָהָם, בַּמַּחֲזֶה, לֵאמֹר: אַל-תִּירָא אַבְרָהָם, אָנֹכִי מָגֵן לָךְ--שְׂכָרְךָ, הַרְבֵּה מְאֹד. וְהֶאֱמִן, בַּיהוָה; וַיַּחְשְׁבֶהָ לּוֹ, צְדָקָה. וַיֹּאמֶר, אֵלָיו: אֲנִי יְהוָה, אֲשֶׁר הוֹצֵאתִיךָ מֵחָרָן--לָתֶת לְךָ אֶת-הָאָרֶץ הַזֹּאת, לְרִשְׁתָּהּ. וַיֹּאמַר: אֲדֹנָי יְהוִה, בַּמָּה אֵדַע כִּי אִירָשֶׁנָּה. וַיֹּאמֶר אֵלָיו, קְחָה לִי עֶגְלָה מְשֻׁלֶּשֶׁת, וְעֵז מְשֻׁלֶּשֶׁת, וְאַיִל מְשֻׁלָּשׁ; וְתֹר, וְגוֹזָל. וַיִּקַּח-לוֹ אֶת-כָּל-אֵלֶּה, וַיְבַתֵּר אֹתָם בַּתָּוֶךְ, וַיִּתֵּן אִישׁ-בִּתְרוֹ, לִקְרַאת רֵעֵהוּ; וְאֶת-הַצִּפֹּר, לֹא בָתָר. וַיֵּרֶד הָעַיִט, עַל-הַפְּגָרִים; וַיַּשֵּׁב אֹתָם, אַבְרָהָם. וַיְהִי הַשֶּׁמֶשׁ לָבוֹא, וְתַרְדֵּמָה נָפְלָה עַל-אַבְרָהָם; וְהִנֵּה אֵימָה חֲשֵׁכָה גְדֹלָה, נֹפֶלֶת עָלָיו. וַיֹּאמֶר לְאַבְרָהָם, יָדֹעַ תֵּדַע כִּי-גֵר יִהְיֶה זַרְעֲךָ בְּאֶרֶץ לֹא לָהֶם, וַעֲבָדוּם, וְעִנּוּ אֹתָם--אַרְבַּע מֵאוֹת, שָׁנָה. וְגַם אֶת-הַגּוֹי אֲשֶׁר יַעֲבֹדוּ, דָּן אָנֹכִי; וְאַחֲרֵי-כֵן יֵצְאוּ, בִּרְכֻשׁ גָּדוֹל. וְאַתָּה תָּבוֹא אֶל-אֲבֹתֶיךָ, בְּשָׁלוֹם: תִּקָּבֵר, בְּשֵׂיבָה טוֹבָה. וְדוֹר רְבִיעִי, יָשׁוּבוּ הֵנָּה: כִּי לֹא-שָׁלֵם עֲוֹן הָאֱמֹרִי, עַד-הֵנָּה.
הַשָּׂדֶה אֲשֶׁר-קָנָה אַבְרָהָם, מֵאֵת בְּנֵי-חֵת--שָׁמָּה קֻבַּר אַבְרָהָם, וְשָׂרָה אִשְׁתּוֹ. וַיִּהְיוּ חַיֵּי שָׂרָה, מֵאָה שָׁנָה וְעֶשְׂרִים שָׁנָה וְשֶׁבַע שָׁנִים--שְׁנֵי, חַיֵּי שָׂרָה. וְאֵלֶּה, יְמֵי שְׁנֵי-חַיֵּי אַבְרָהָם--אֲשֶׁר-חָי מְאַת שָׁנָה וְשִׁבְעִים שָׁנָה, וְחָמֵשׁ שָׁנִים, וַיִּגְוַע. וַיִּקְבְּרוּ אֹתוֹ יִצְחָק וְיִשְׁמָעֵאל, בָּנָיו, אֶל-מְעָרַת, הַמַּכְפֵּלָה: אֶל-שְׂדֵה עֶפְרֹן בֶּן-צֹחַר, הַחִתִּי, אֲשֶׁר, עַל-פְּנֵי מַמְרֵא. וַיָּקָם הַשָּׂדֶה וְהַמְּעָרָה אֲשֶׁר-בּוֹ, לְאַבְרָהָם--לַאֲחֻזַּת-קָבֶר: מֵאֵת, בְּנֵי-חֵת.
וַיְהִי אַחֲרֵי מוֹת אַבְרָהָם וַיְבָרֶךְ אֱלֹהִים אֶת יִצְחָק בְּנוֹ וַיֵּשֶׁב יִצְחָק, עִם-בְּאֵר לַחַי רֹאִי. וַיִּשְׁכְּן יִשְׁמָעֵאל מֵחֲוִילָה עַד-שׁוּר, אֲשֶׁר עַל-פְּנֵי מִצְרַיִם, בֹּאֲכָה, אַשּׁוּרָה; עַל-פְּנֵי כָל-אֶחָיו ,נָפָל. וְאֵלֶּה, שְׁנֵי חַיֵּי יִשְׁמָעֵאל--מְאַת שָׁנָה וּשְׁלֹשִׁים שָׁנָה, וְשֶׁבַע שָׁנִים; וַיִּגְוַע. וַיֶּעְתַּר יִצְחָק לַיהוָה לְנֹכַח אִשְׁתּוֹ, כִּי עֲקָרָה הִוא; וַיֵּעָתֶר לוֹ יְהוָה, וַתַּהַר רִבְקָה אִשְׁתּוֹ. וַיִּתְרֹצְצוּ הַבָּנִים, בְּקִרְבָּהּ, וַתֹּאמֶר אִם-כֵּן, לָמָּה זֶּה אָנֹכִי; וַתֵּלֶךְ, לִדְרֹשׁ אֶת-יְהוָה. וַיֹּאמֶר יְהוָה לָהּ, שְׁנֵי גֹיִים בְּבִטְנֵךְ, וּשְׁנֵי לְאֻמִּים, מִמֵּעַיִךְ יִפָּרֵדוּ; וּלְאֹם מִלְאֹם יֶאֱמָץ, וְרַב יַעֲבֹד צָעִיר. וַיִּמְלְאוּ יָמֶיהָ, לָלֶדֶת; וְהִנֵּה תוֹמִם, בְּבִטְנָהּ. וַיֵּצֵא הָרִאשׁוֹן אַדְמוֹנִי, כֻּלּוֹ כְּאַדֶּרֶת שֵׂעָר; וַיִּקְרְאוּ שְׁמוֹ, עֵשָׂו. וְאַחֲרֵי-כֵן יָצָא אָחִיו, וְיָדוֹ אֹחֶזֶת בַּעֲקֵב עֵשָׂו, וַיִּקְרָא שְׁמוֹ, יַעֲקֹב; וְיִצְחָק בֶּן-שִׁשִּׁים שָׁנָה, בְּלֶדֶת אֹתָם. וַיִּהְיוּ, יְמֵי יִצְחָק--מְאַת שָׁנָה, וּשְׁמֹנִים שָׁנָה, וַיִּגְוַע. וַיִּקְבְּרוּ אֹתוֹ, עֵשָׂו וְיַעֲקֹב בָּנָיו.
וַיֶּאֱהַב יַעֲקֹב, אֶת-רָחֵל; וַיַּעֲבֹד יַעֲקֹב בְּרָחֵל, שֶׁבַע שָׁנִים; וַיִּהְיוּ בְעֵינָיו כְּיָמִים אֲחָדִים, בְּאַהֲבָתוֹ אֹתָהּ. וַיִּזְכֹּר אֱלֹהִים, אֶת-רָחֵל; וַיִּשְׁמַע אֵלֶיהָ אֱלֹהִים, וַיִּפְתַּח אֶת-רַחְמָהּ. וַתַּהַר, וַתֵּלֶד בֵּן; וַתֹּאמֶר, אָסַף אֱלֹהִים אֶת-חֶרְפָּתִי. וַתִּקְרָא אֶת-שְׁמוֹ יוֹסֵף. וַיָּבֵא יוֹסֵף אֶת-יַעֲקֹב אָבִיו, וַיַּעֲמִדֵהוּ לִפְנֵי פַרְעֹה; וַיְבָרֶךְ יַעֲקֹב, אֶת-פַּרְעֹה. וַיֹּאמֶר פַּרְעֹה, אֶל-יַעֲקֹב: כַּמָּה, יְמֵי שְׁנֵי חַיֶּיךָ. וַיֹּאמֶר יַעֲקֹב, אֶל-פַּרְעֹה, יְמֵי שְׁנֵי מְגוּרַי, שְׁלֹשִׁים וּמְאַת שָׁנָה: מְעַט וְרָעִים, הָיוּ יְמֵי שְׁנֵי חַיַּי, וְלֹא הִשִּׂיגוּ אֶת-יְמֵי שְׁנֵי חַיֵּי אֲבֹתַי, בִּימֵי מְגוּרֵיהֶם. וַיְבָרֶךְ יַעֲקֹב, אֶת-פַּרְעֹה; וַיֵּצֵא, מִלִּפְנֵי פַרְעֹה. וַיּוֹשֵׁב יוֹסֵף, אֶת-אָבִיו וְאֶת-אֶחָיו, וַיִּתֵּן לָהֶם אֲחֻזָּה בְּאֶרֶץ מִצְרַיִם, בְּמֵיטַב הָאָרֶץ בְּאֶרֶץ רַעְמְסֵס. וַיְכַלְכֵּל יוֹסֵף אֶת-אָבִיו וְאֶת-אֶחָיו, וְאֵת כָּל-בֵּית אָבִיו--לֶחֶם, לְפִי הַטָּף. ויִּפְרוּ וַיִּרְבּוּ מְאֹד. וַיְחִי יַעֲקֹב בְּאֶרֶץ מִצְרַיִם, שְׁבַע עֶשְׂרֵה שָׁנָה; וַיְהִי יְמֵי-יַעֲקֹב, שְׁנֵי חַיָּיו--שֶׁבַע שָׁנִים, וְאַרְבָּעִים וּמְאַת שָׁנָה, וַיִּגְוַע
(Download here)
Sunday, March 20, 2016
Purim Sameach
We associated two very different stories to Purim, the story of the Megillah of Esther, and the story of the Aramaic "Megillah" of Ezra 7. Both are "stories" of how the king of Paras treated the Jews.
And both stories are about the beginning of the world of Torah. There seems to be another connection between the two stories.
The first story is one of crisis, unity, and survival. The second story features implicitly the self-aggrandizement of the Kohanim, and the associated pious fraud by the redactor. Strikingly, HaShem is not mentioned in the Megillah of Esther. He hides Himself in the plot of the story. The second Megillah is filled with references to G-d.
The connection is the following. The Truth, humble and straight, is hidden by a layer of lies, self-aggrandizement and pious fraud. The message of Purim is that the Truth is present, in the plot of reality as in the Torah, to who can look through the obfuscating layer. On Purim, look for the Truth and you will find. Purim Sameach!
The first story is one of crisis, unity, and survival. The second story features implicitly the self-aggrandizement of the Kohanim, and the associated pious fraud by the redactor. Strikingly, HaShem is not mentioned in the Megillah of Esther. He hides Himself in the plot of the story. The second Megillah is filled with references to G-d.
The connection is the following. The Truth, humble and straight, is hidden by a layer of lies, self-aggrandizement and pious fraud. The message of Purim is that the Truth is present, in the plot of reality as in the Torah, to who can look through the obfuscating layer. On Purim, look for the Truth and you will find. Purim Sameach!
Friday, March 18, 2016
Possible Scroll of PE
The word of Hashem came to Avraham in a vision saying, "Fear not, Avraham, I am a shield for you; your reward is very great." And he trusted in Hashem, and He reckoned it to him as righteousness. He said to him, "I am Hashem Who brought you out of Haran to give you this land to inherit it." He said, "My Lord, Hashem - Whereby shall I know that I am to inherit it?" And He said to him, "Take to Me three heifers, three goats, three rams, a turtledove, and a young dove." He took all these to Him - he cut them in the center, and placed each piece opposite its counterpart. The birds, however, he did not cut up. Birds of prey descended upon the carcasses, and Avraham drove them away. And it happened, as the sun was about to set, a deep sleep fell upon Avraham; and behold — a dread! great darkness fell upon him. And He said to Avraham, "Know with certainty that your offspring shall be aliens in a land not their own, they will serve them, and they will oppress them four hundred years. But also the nation that they shall serve, I shall judge, and afterwards they shall leave with great wealth. As for you - you shall come to your ancestors in peace; you shall be buried in a good old age. And the fourth generation shall return here, for the iniquity of the Amorite shall not yet be full until then."
And Sarah's life was a hundred years and twenty years and seven years: the years of Sarah's life. And these are the days of the years of Avraham's life that he lived: a hundred years and seventy years and five years, and he expired. And Yitzchak and Yishmael, his sons, buried him at the cave of Machpelah at the field of Ephron, son of Zohar, the Hittite, facing Mamre, the field that Avraham had bought from the children of Heth. Avraham was buried there -- and Sarah, his wife. And the field and the cave that was in it were established for Avraham as a posession for a tomb from the children of Heth. And it was after Abraham's death, and God blessed Yitzchak his son. And Yitzchak settled near Beer-lahai-roi. And Yishmael dwelt from Havilah to Shur — which is near Egypt — toward Assyria; he fell facing all his brothers. And these are the years of Yishmael's life: a hundred years and thirty years and seventy years, and he expired.
Yitzchak entreated Hashem opposite his wife, because she was barren. Hashem allowed Himself to be entreated by him, and his wife Rivka conceived. The children agitated within her, and she said, "If so, why am I thus?" And she went to inquire of Hashem. And Hashem said to her, "Two nations are in your womb; two regimes from your insides shall be separated; the might shall pass from one regime to the other, and the elder shall serve the younger." When her term to bear grew full, then behold! there were twins in her womb. The first one emerged red, entirely like a hairy mantle; so they named him Esav. After that his brother emerged with his hand grasping on to the heel of Esav; so he called his name Ya'akov; Yitzchak was sixty years old when she bore them. And Yitzchak's days were a hundred years and eighty years, and he expired. And Esav and Ya'akov, his sons, buried him.
And Jacob worked seven years for Rachel and they seemed to him a few days because of his love for her. God remembered Rachel; God hearkened to her and He opened her womb. She conceived and bore a son, and said, "God has taken away my disgrace." So she called his name Yoseph. And Yoseph brought Ya'akov, his father, and stood him in front of Par'oh. And Ya'akov blessed Par'oh. And Par'oh said to Ya'akov, "How many are the days of the years of your life?" And Ya'akov said to Par'oh, "The days of the years of my residences are a hundred thirty years. The days of the years of my life have been few and bad, and they haven't attained the days of the years of my father's lives, in the days of their residences." And Ya'akov blessed Par'oh and went out from in front of Par'oh. And Yoseph settled his father and his brothers and gave them a possession in the land of Mitzraim, in the best of the land, in the land of Rameses. And Yoseph supported his father and his brothers and all of his father's household, bread by the number of infants. And they were fruitful and multiplied very much. And Ya'akov lived in the land of Mitzraim seventeen years. And Ya'akov's days, the years of his life, were seven years and a hundred forty years, and he expired.
And Sarah's life was a hundred years and twenty years and seven years: the years of Sarah's life. And these are the days of the years of Avraham's life that he lived: a hundred years and seventy years and five years, and he expired. And Yitzchak and Yishmael, his sons, buried him at the cave of Machpelah at the field of Ephron, son of Zohar, the Hittite, facing Mamre, the field that Avraham had bought from the children of Heth. Avraham was buried there -- and Sarah, his wife. And the field and the cave that was in it were established for Avraham as a posession for a tomb from the children of Heth. And it was after Abraham's death, and God blessed Yitzchak his son. And Yitzchak settled near Beer-lahai-roi. And Yishmael dwelt from Havilah to Shur — which is near Egypt — toward Assyria; he fell facing all his brothers. And these are the years of Yishmael's life: a hundred years and thirty years and seventy years, and he expired.
Yitzchak entreated Hashem opposite his wife, because she was barren. Hashem allowed Himself to be entreated by him, and his wife Rivka conceived. The children agitated within her, and she said, "If so, why am I thus?" And she went to inquire of Hashem. And Hashem said to her, "Two nations are in your womb; two regimes from your insides shall be separated; the might shall pass from one regime to the other, and the elder shall serve the younger." When her term to bear grew full, then behold! there were twins in her womb. The first one emerged red, entirely like a hairy mantle; so they named him Esav. After that his brother emerged with his hand grasping on to the heel of Esav; so he called his name Ya'akov; Yitzchak was sixty years old when she bore them. And Yitzchak's days were a hundred years and eighty years, and he expired. And Esav and Ya'akov, his sons, buried him.
And Jacob worked seven years for Rachel and they seemed to him a few days because of his love for her. God remembered Rachel; God hearkened to her and He opened her womb. She conceived and bore a son, and said, "God has taken away my disgrace." So she called his name Yoseph. And Yoseph brought Ya'akov, his father, and stood him in front of Par'oh. And Ya'akov blessed Par'oh. And Par'oh said to Ya'akov, "How many are the days of the years of your life?" And Ya'akov said to Par'oh, "The days of the years of my residences are a hundred thirty years. The days of the years of my life have been few and bad, and they haven't attained the days of the years of my father's lives, in the days of their residences." And Ya'akov blessed Par'oh and went out from in front of Par'oh. And Yoseph settled his father and his brothers and gave them a possession in the land of Mitzraim, in the best of the land, in the land of Rameses. And Yoseph supported his father and his brothers and all of his father's household, bread by the number of infants. And they were fruitful and multiplied very much. And Ya'akov lived in the land of Mitzraim seventeen years. And Ya'akov's days, the years of his life, were seven years and a hundred forty years, and he expired.
Thursday, March 17, 2016
Strands of JE in PE
It was natural to combine the text of PE with strands of JE, giving the background of the text and containing Bereshit 15:13 and Bereshit 25:26,
as explained here. These strands of JE became part of PE, and later of the P scroll. When the redactor combined P with JE, these strands were readily recognized. This can explain the apparent gap in P regarding the birth of Ya'akov and Esav. It was in the P scroll, but was a strand of JE. The next post is a possible combination of PE with strands of JE. The bold Pasuk would have been added by PE to JE, an introduction to the text that describes the life of Yitzchak. P has two such introductions:
25:19: "Avraham begot Yitzchak. And Yitzchak was ..."
25:13: "And it was after Avraham death and G-d blessed Yitzchak, his son."
It makes sense that one of the two is from PE.
25:19: "Avraham begot Yitzchak. And Yitzchak was ..."
25:13: "And it was after Avraham death and G-d blessed Yitzchak, his son."
It makes sense that one of the two is from PE.
Tuesday, March 15, 2016
The Mathematics of PE
Let us have a short look at the mathematics of PE. It uses the set of functions Aa(n)
Aa(n) = (a-2n)n2 = an2 - 2n3
One readily sees that
A17(5) = 175
A17(6) = 180
A17(7) = 147
are the ages of Avraham, Yitzchak, and Ya'akov. The implied age is
A17(4) = 144
and is the only one left that is over 100. The ages
A17(8) = 64
A17(3) = 99
will be explained later. The maximum age is reached when
2an - 6n2 = 0
which is at n = a/3, for continuous n, and the maximum reaches the value
Aa(a/3) = (a/3)3
For a=17, the value a/3 is closest to 6, and therefore Yitzchak reaches the oldest age.
Aa(n) = (a-2n)n2 = an2 - 2n3
One readily sees that
A17(5) = 175
A17(6) = 180
A17(7) = 147
are the ages of Avraham, Yitzchak, and Ya'akov. The implied age is
A17(4) = 144
and is the only one left that is over 100. The ages
A17(8) = 64
A17(3) = 99
will be explained later. The maximum age is reached when
2an - 6n2 = 0
which is at n = a/3, for continuous n, and the maximum reaches the value
Aa(a/3) = (a/3)3
For a=17, the value a/3 is closest to 6, and therefore Yitzchak reaches the oldest age.
Monday, March 14, 2016
The PE Text is Divine
A scribe created a text relating the ages that were reached by the Avot. The text refers indirectly to the numbers 144 and 210, and so refers to all the numbers 127, 274/2, 288/2, 351/2, 420/2, 703/4. The scribe had no computers without which we would not know the equations:
10! + 152 = (15*127)2
14! + 4202 = (420*703)2
11! + 182 = (18*351)2
13! + 2882 = (288*274)2
The PE text, of the scribe, is Divine.
10! + 152 = (15*127)2
14! + 4202 = (420*703)2
11! + 182 = (18*351)2
13! + 2882 = (288*274)2
The PE text, of the scribe, is Divine.
Sunday, March 13, 2016
The Numbers of Ayil Meshulash
So, as one half of 351 is 175.5 (and one fourth of 703 is 175.75), it points (they point) to the age of Avraham. The latter is rounded as 175, and not as 176, to enable the Derasha of PE, the mathematical series hidden in the sequence 175, 180, 147, the ages of the Avot, which leads to 144.
And so we have 127, 137, 144, 175, 210 as numbers that are equal to (half) the numbers of Ayil Meshulash:
10! + 152 = (15*127)2
14! + 4202 = (420*703)2
11! + 182 = (18*351)2
13! + 2882 = (288*274)2
Conversely, the numbers 127, 274/2, 288/2, 351/2, 420/2, 703/4 are numbers of PE. This is mathematically impressive.
10! + 152 = (15*127)2
14! + 4202 = (420*703)2
11! + 182 = (18*351)2
13! + 2882 = (288*274)2
Conversely, the numbers 127, 274/2, 288/2, 351/2, 420/2, 703/4 are numbers of PE. This is mathematically impressive.
The Intention Of PE
One may wonder about the number 130 in PE (Bereshit 47:9):
And Ya'akov said to Par'oh, "The days of the years of my residences are a hundred thirty years.
What is its purpose? It is the only number in PE which appears to have no relationship with a number in Ayil Meshulash. The answer must be that the purpose is the connection between J, Bereshit 25:26, which says that Yitzchak was 60 years old when his sons were born, and E, Bereshit 15:13, which says that the Avraham's seed will be degraded 400 years. It follows from these numbers that Galut Mitzrayim lasted 210 years, one half of the only number that PE did not predict yet, 420. There is a relationship with a number in Ayil Meshulash after all. The age of Ya'akov was 130 when the 210 years started. The above would seem to imply that the intention of PE was to be combined with EJ, as we did.
And Ya'akov said to Par'oh, "The days of the years of my residences are a hundred thirty years.
What is its purpose? It is the only number in PE which appears to have no relationship with a number in Ayil Meshulash. The answer must be that the purpose is the connection between J, Bereshit 25:26, which says that Yitzchak was 60 years old when his sons were born, and E, Bereshit 15:13, which says that the Avraham's seed will be degraded 400 years. It follows from these numbers that Galut Mitzrayim lasted 210 years, one half of the only number that PE did not predict yet, 420. There is a relationship with a number in Ayil Meshulash after all. The age of Ya'akov was 130 when the 210 years started. The above would seem to imply that the intention of PE was to be combined with EJ, as we did.
Friday, March 4, 2016
V'nahafoch Hu
The Torah of the redactor obviously met with extremely serious opposition. Here the return to the Land of Israel and the authority of the king, known from the book of Esther, colluded. The king of Persia was impressed by the Torah of G-d, in the hands of Ezra, see Ezra 7. He gave Ezra the worldly power to crush his opposition.
We celebrate on Purim the failed physical destruction of Israel by the hands of the Persian king, the enemy being given into our hands. But the spiritual destruction that the Persian king did accomplish, giving the power of king of Israel into the hands of the Kohanim, is still with us today. V'nahafoch Hu.
We celebrate on Purim the failed physical destruction of Israel by the hands of the Persian king, the enemy being given into our hands. But the spiritual destruction that the Persian king did accomplish, giving the power of king of Israel into the hands of the Kohanim, is still with us today. V'nahafoch Hu.
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