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We find a reference to a triple repetition of terms in Parashat Beshalach (Exodus 15:18), where it states, “Hashem yimloch le’olam va’ed ? G-d will reign forever and ever.”
Onkelos, in his Aramaic commentary-augmented translation, states, “Hashem malchutei ka’im le’alam u’le’almei almaya,” which is translated as “The reign of G-d is eternal, forever and ever.” We thus find in his translation a three-fold repetition of the word alam to express everlasting eternity.
Indeed, that is how Rambam (Hilchot Sanhedrin 18:5) understands it, stating, “If he yet again, a third time, violated a karet prohibition and was warned, he is condemned to be fed barley bread until he expires.” Rambam stresses that he is given three warnings, and only after the third unheeded violation is capital punishment applicable.
We find another instance (Yoreh De’ah 228:3) of repeating something thrice: “hatara,” the nullification of vows. The Beit Din nullifies vows by pronouncing three times either “mutar lach” (it is permissible to you), “sharei lach” (a similar meaning), or “machul lach” (it is forgiven to you).
The Shach (ad loc.) explains the three times as being solely for the purpose of underscoring that point, but that as a matter of strict halacha, once would suffice.
The Talmud (Yoma 85b, Mishna) states “… [regarding] sins between man and his fellow man, Yom Hakippurim does not atone until he asks his fellow’s forgiveness.”
The Gemara (87a) then quotes R. Hisda, who requires the sinner to ask forgiveness before three groups of three people each. R. Yosi b. Hanina states that whoever asks forgiveness of his fellow man should not do so more than three times (if the latter remains unappeased).
Indeed, the Mechaber (Orach Chayyim 606:1) rules accordingly. He teaches that after three such pleadings which have been ignored, the offender no longer bears any iniquity, and Yom Kippur will surely atone.
There are other reasons for repeating Shalom Aleichem three times during Kiddush Levana. Bnei Yissas’char (Ma’amarim 4 and 5, Kiddush Hachodesh) states: “According to the holy words of the Arizal, we say Shalom Aleichem three times after reciting birkat ha’levana because the very first kitrug (denouncement, which is exactly the opposite of shalom, harmony) was caused by the moon, which said (Chullin 60b): ‘It is impossible for two kings to wear one crown.’ The moon was then ordered to diminish itself in size. As a consolation, the Gemara states that G-d told the moon that righteous men shall be named in reference to the moon, the small luminary (hama’or hakatan). Thus we find that our Patriarch Jacob is called katan (Amos 7:2, referring to the Jewish nation), we have Shemuel hakatan (the Tanna Shemuel), and David (I Samuel 17:14), who was the katan among his brothers. This is another reference to the number three.”
“Now our blessing for the moon is that the blemish (in its light) should be repaired so that it will be restored to its wholeness, and thus the light of the moon will be like the light of the sun, resulting in a restoration of harmony. Thus, as the prophet Isaiah states (11:6), “Vegar ze’ev im keves ve’namer im gedi yirbatz - The wolf shall dwell with the sheep and the leopard with the kid.” Therefore, at the blessing of the moon, we say to each other ‘Shalom Aleichem – peace be unto you.’” This concludes the Arizal’s statement, as discussed by Bnei Yissas’char.
Bnei Yissas’char then asks: “Why do we specifically recite the verses three times? It is written (Psalms 119:165): ‘Shalom rav le’ohavei toratecha ve’ein lamo michshol - Manyfold peace to the lovers of Your Torah; they shall not encounter any stumbling blocks.’ This is in accord with
what our sages (Gittin 46a) state: ‘yamim’ (days) refers to two, whereas the term ‘rabbim’ (many) refers to three. Therefore the use of ‘ribbuy,’ the plural count, refers to three. The verse actually means that when shalom (harmony) is recited many times, [at least] three, no stumbling block will be encountered.”
Therefore, at the monthly renewal of the moon, we say Shalom Aleichem three times in order that there be no stumbling block for us during the new month.
Bnei Yissas’char points out the three names of the moon found in the Tanach: yare’ach, levana, and sahar. This too serves as a reason to recite Shalom Aleichem three times, once for each of the moon’s names.
We find yet another explanation in Bnei Yissas’char (Ma’amar Kislev 13, as well as in our ma’amar). He mentions the three fundamental mitzvot: Chodesh Kiddush Hachodesh, through which we count our months and proclaim our festivals), Shabbat and milah (all the mitzvot whose observance the Syrian Greeks attempted to void and thus eliminate all Torah observance – see Megillat Antiochus).
Thus these three mitzvot serve as yet another reason for our saying “Shalom Aleichem” three times.
We can see that there are numerous reasons for our repeating these special phrases three times. We do so every month, as we await our final deliverance, speedily in our days.