"And He appeared to him" - The word "And He appeared" indicates that God revealed Himself to Abraham for the purpose of the word "DA" (which means "knowledge" in Hebrew), which is spelled out by the first letters of the words "Nimlo Atto" (meaning "circumcised with him"). And the word "Samuch" (meaning "near") teaches that God appeared to him because he was sitting at the entrance of his tent on a hot day. This is why our sages said that Abraham was sitting at the entrance of hell and did not allow anyone who was circumcised to enter, except for those who converted to Judaism and were circumcised because their foreskin was pulled back and they were recognized as having been properly circumcised.

The phrase "'וירא אליו ה" with the letter 'ה in the middle (a notation indicating that this is how it appears in a Torah scroll) teaches that the Divine Presence ('ה) came to dwell in the tent of Abraham.

"And He appeared to him" with the letters' numerical value (gematria), this implies that the word appeared (זה) was revealed to him with great honor.

"And the Lord appeared to him in the oaks of Mamre" - The gematria of "in the oaks of Mamre" is that He came to visit the sick.

The phrase "יושב פתח האהל" (sitting at the entrance of the tent) has the numerical value of "to welcome guests."

The word "כחם" in the Scripture means "like the heat of the day," like the heat of the sun, or like the clear, bright light. "And they came like the heat of the day to the house of the man of shame" (Jeremiah 15:8). Here, "like the heat of the day" means not the scorching heat of the sun, but rather the bright and clear light of the Garden of Eden, as it is written, "And the Lord shall be unto thee an everlasting light" (Isaiah 60:19). However, "the man of shame" means someone who came through conversion for ulterior motives, for there is no greater shame than one who comes through conversion. "Like the heat of the sun" refers to someone who will go to Gehenna, as it is written, "Behold, the day cometh, it burns like a furnace" (Malachi 3:19). "כחם היום" (like the heat of the day) in gematria is equivalent to "זהו בגיהנם" (this is in Gehenna), and in gematria it is equivalent to the word "דם" (blood).

"והנה שלשה" [And here are three] in gematria refers to Michael, Gabriel, and Raphael.

"Let a little water be brought" - here it says "three" [men], and elsewhere [in the same verse] it says, "and let a morsel of bread be eaten." The explanation is that one of the three is a captive, and a sword of destruction is taken from him. The righteous captive is taken from among the wicked, in the merit of the words "Please take a little water." "Iron from the dust is taken" - this means that the sword of destruction is taken from Israel, who are compared to dust, in the merit of Abraham, who said, "I am but dust and ashes." "Please take a little water" has the same numerical value as "from here they merited the well."

"And bathe your feet" - here it says "two," and elsewhere [in the same verse] it speaks of Lot. This teaches that Lot was afraid of robbers, so he preferred to spend the night in the city rather than outside. Abraham, on the other hand, chose to bathe first, indicating that he was not afraid of robbers and would welcome any traveler into his home.

"And I will fetch a morsel of bread" - It would have been sufficient for him to say, "Take a morsel of bread." From here we learn that the host should be proactive.

Your hearts [לבכם] - three [words] in the Tanakh here and there [in the text]. [It means] "set your heart to worry." "וראיתם ושש לבכם" - "and you shall see, and your heart will be glad." This is [to teach us] that [Abraham] was pained that guests had not come to him, and this is [why it says] "set your heart to worry," that he was worried in his heart when he saw them, and "your heart will be glad," as his heart was gladdened [when they arrived].

"and to the cattle" - The letter "ק" (Kuf) is written small to indicate that there were only three animals. In the merit of this, he was granted a son at the age of one hundred.

"And he ran to the cattle" - The letters of "El HaKever" (to the burial place) are rearranged to form the word "Ratz" (he ran). He ran after them to the cave. "And he ran to the cattle" - The gematria (numerical value) of "Ratz LaMearah" (he ran to the cave) is equal to the gematria of "El HaBakar" (to the cattle).

"And they ate" and later they said to him (Abraham), this is a hint to what they said that we do not speak about the Messiah during a meal, lest our trachea twist in front of our esophagus (and make us choke).

The phrase "ויאמרו אליו" (vayomeru elav) in Genesis 18:9 has a numerical value (gematria) of 241, which is equal to the phrase "אמרו אליה איו" (amaru eleha ayu), meaning "they said to her, 'Where is he?'"

"the way of women, written in two ways, Rosh Tevot and Sefer Torah, equals the value of the word 'niddah' in gematria."

"My lord is old" has the numerical value of "grinding but not releasing."

"On the face of Sodom" - the letters "samech, tet" (ס״ת) have the numerical value of "mil" (מיל), indicating that the measurement of a mil (distance) went with them to Sodom. This is to teach us that the angel who accompanied Lot measured a mil's worth of distance with him to escort him out of the city.

"Behold" go with them and the Lord. This teaches that the Shekina accompanies the righteous."

"And Abraham" the letter "Heh" is at the beginning of the verse, "And Abraham and Sarah were old." Abraham will be, meaning even though Abraham and Sarah were old, Abraham will become a great nation. Abraham was old, and he was 99 years old. Abraham was a hundred years old. This means that despite the fact that Abraham and Sarah were old, Abraham would become a great nation.

The phrase "היו" (will be) has a gematria value of 21. The interpretation is that when one is at the age of 21 (בן ק"ן), he will become a great nation. This aligns with the idea expressed in the verse "סי כן יבורך גבר" (The one who diligently seeks good will be blessed).

"Who will command his children" – The numerical value(612) of the letters of "אשר יצוה" (who will command) is equal to the numerical value of the word "תורה" (Torah), and in the numerical value of "ברית" (covenant).

"For 'Rabah' [is] in gematria 'sin' multiplied."

"The righteous, the righteous, four against four. Judges who were in Sodom, and all of them lacked a measure of righteousness."

"And I am dust and ashes. From here, [Abraham] merited the ashes of the Red Heifer and the dust of the Sotah (a woman suspected of adultery). 'Dust' in gematria (numerical value) is equal to 'Sotah' with a left Shin (ש), and 'Ashes' in gematria is equal to 'Parah' (Heifer). Dust and ashes. In one context here [Genesis], and in another in the Book of Job. And I will be comforted over dust and ashes. This is greater than what is said in Job than what is said about Abraham."