Who knows fifty-seven?

Please cite/link your sources, if possible. At some point in the next few days, I will:

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There are 57 b'rachos that are "in the bag" when calculating a [chol day] calculation of 100 b'rachos, due to the 3 daily amidos (19*3=57).

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57 are the number of "houses" (major sections) in R' Yaakov Emden's Siddur Ammudei Shamayim, covering the weekday and Shabbos prayers. Most are named for parts of the Beis Hamikdash or of Shlomo's palace.

57 is also the gematria of the word זן ("feeds"), the theme of the first blessing of Birkas Hamazon (called for this reason ברכת הזן). Corresponding to this, there are 57 words in Psalm 23, and there are Kabbalistic sources that say that it is appropriate to recite this before the meal as a segulah for a good livelihood. R' Yaakov Emden also comments that it is the gematria of the two Divine names א-ל הוי"ה, associated with G-d's provision of food for us, and indeed we invoke Him by these two names in Birkas Hamazon. (In Nusach Ashkenaz, too, ברכת הזן itself contains 57 words.)

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From Taame Haminhagim, kuntres acharon 51 to paragraph 320 (translation my own, possibly poor):

One can say the reason [for saying וינוחו בה, בו, בם, respectively, in the evening, day, and evening prayers of Shabas] is that in a standard year there are fifty Shabasos and seven "mikra kodesh" holidays from the Torah (two of Pesach, two of Sukos, Shavuos, Rosh Hashana, and Yom Kipur), totaling fifty-seven days of rest. The words בה, בו, בם total fifty-seven in gimatriya; they hinted with these to the fifty-seven days of the year — namely, the Shabasos and holidays — on which "ינוחו... ישראל מקדשי שמך".‎

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