And the messengers returned to Jacob, saying: “We came to thy brother Esau, and moreover he cometh to meet thee, and four hundred men with him.” (Genesis 32:7)

Why does the Torah take pains to tell us the exact number of men accompanying Esau? Servants of Esau are of no particular significance. It is apparent that, by specifying the number of men in Esau’s entourage, the Torah is trying to tell us something about Esau.

It seems to me that Jacob understood the significance of the number 400. This number is a sum of the powers of 7 from 0 to 3: 70+71+72+73=400.

These four powers of 7 hint at the lunar month, which has approximately 28 days (4×7). The significance of this will become apparent later. 400 is a repdigit (a portmanteau of repeated and digit—a natural number composed of repeated instances of the same digit in a positional number system) in base 7 (1111). Esau and his 400 men together were 401 strong—and 401 is a prime number that is the sum of seven consecutive prime numbers (43+47+53+59+61+67+71). It seems to be all about the number 7! Jacob may have alluded to this by bowing down to his brother seven times as he approached Esau, as the Torah says:

And he himself passed over before them, and bowed himself to the ground seven times, until he came near to his brother. (Genesis 33:3)

More significantly, though, the number 400 is an important number in the astronomy of the solar system. If you have ever watch sunset long enough, you might have noticed that just before the sun drops below the horizon and as the moon rises in the night sky, they appear to be roughly the same size. How could that be? Isn’t the sun much bigger than the moon? You have the number 400 to blame!

Indeed, the diameter of the sun is roughly 400 times larger than the diameter of the moon. The sun is also about 400 times further away from Earth than the moon is.  (It is this coincidence that makes total solar eclipses possible.) Hence the two appear to be about the same size. This relationship in sizes was already well understood in antiquity by Aristarchus (310–230 B.C.E.), although he erred as to the actual sizes and distances.

What significance, however, does the number 400 have to the Biblical story of Jacob and Esau?

Just as the moon and the sun appear to be of the same size, Jacob and Esau appear to be twin brothers. In reality, however, just as the moon and the sun are very different in size and in nature (the former being a satellite of a planet and the latter being a star), so too, there is little in common between Jacob and Esau, besides that they are brothers. As I wrote in my essay, The Entangled Twins, G‑d’s choice of Jacob over Esau is a metaphor for spontaneous symmetry breaking:

I loved you, said the Lord, and you said, “How have You loved us?” Was not Esau a brother to Jacob? says the Lord. And I loved Jacob. And I hated Esau. (Malachi 1, 2-3)

The Talmud compares the Jewish people—the descendants of Jacob—to the moon, whereas it compares gentiles—the descendants of Esau—to the sun: “…the gentiles reckon their calendar by the Sun and Israel by the Moon.” (Babylonian Talmud, tr. Sukkah 2:9) While, historically, the Jewish people’s fortune has waxed and waned, the moon is ultimately the symbol of rebirth and renewal.

Let us now summarize the hints contained in this story from the point of view of the number-theoretical allegory:

The story of the meeting between Jacob and Esau is dominated by number 7—a prime number. The number of Esau’s entourage, 400, is the sum of four powers of 7 from 0 to 3. These four powers of 7 hint at the lunar month (4 × 7 = 28), by which Jacob’s descendants recon time. Moreover, Esau and his 400 men—401 strong—is a prime number that is the sum of seven consecutive prime numbers. The number 7 is further underscored by seven times that Jacob bowed down to Esau.

As discussed above, the soul of Esau comes from the universe of Tohu—an allegory of prime numbers. The soul of Jacom comes from the universe of Tikkun—an allegory of composite numbers. Tohu is loftier than Tikkun and precedes it in the causal order of the creation. The root-source of Esau’s soul is higher than the root-sours of Jacob’s soul. (That is why Isaac wanted to bless Esau to help him realize his potential.) Jacob recognizes the supremacy of Esau’s spiritual source in Tohu and admits it by bowing down before him seven times.

Lastly, the number 400—the number of Esau’s entourage and the factor of the relative diameter sizes and the respective distances from Earth between the sun and the moon that makes them look the same size—is a metaphor for the apparent similarity of Esau and Jacob, as twins, that in reality are very different indeed.