# 153 (number)

 ← 152 153 154 →
Cardinal one hundred fifty-three
Ordinal 153rd
(one hundred and fifty-third)
Factorization 32× 17
Divisors 1, 3, 9, 17, 51, 153
Roman numeral CLIII
Binary 100110012
Ternary 122003
Quaternary 21214
Quinary 11035
Senary 4136
Octal 2318
Duodecimal 10912
Vigesimal 7D20
Base 36 4936

153 (one hundred [and] fifty-three) is the natural number following 152 and preceding 154.

## Mathematical properties

The number 153 is the 17th triangular number. The colours show that 153 is also the sum of the first five positive factorials.

The number 153 has several interesting mathematical properties. 153 is the sum of the first 17 integers and is also the sum of the first five positive factorials, 1! + 2! + 3! + 4! + 5!. The number 153 is associated with the geometric shape known as the Vesica Piscis or Mandorla. Archimedes, in his Measurement of a Circle, referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of 1/√3.

As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:${\displaystyle 1!+2!+3!+4!+5!}$.[1]

The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.

The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.

Since ${\displaystyle 153=1^{3}+5^{3}+3^{3}}$, it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits.[2] Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407.[3] It is also a Friedman number, since 153 = 3 × 51, and a Harshad number in base 10, being divisible by the sum of its own digits.

The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.

Another interesting feature of the number 153 is that it is the limit of the following algorithm:[4][5]

1. Take a random positive integer, divisible by three.
2. Split that number into its base 10 digits.
3. Take the sum of their cubes.
4. Go back to the second step.

An example, starting with the number 84:

{\displaystyle {\begin{aligned}8^{3}+4^{3}&=&512+64&=&576\\5^{3}+7^{3}+6^{3}&=&125+343+216&=&684\\6^{3}+8^{3}+4^{3}&=&216+512+64&=&792\\7^{3}+9^{3}+2^{3}&=&343+729+8&=&1080\\1^{3}+0^{3}+8^{3}+0^{3}&=&1+0+512+0&=&513\\5^{3}+1^{3}+3^{3}&=&125+1+27&=&153\\1^{3}+5^{3}+3^{3}&=&1+125+27&=&153\end{aligned}}}

## In the Bible

The Miraculous catch of 153 fish by Duccio, 14th century, showing Jesus and the 7 fishing disciples (with Saint Peter leaving the boat).

The Gospel of John (chapter 21:1–14) includes the narrative of the miraculous catch of 153 fish as the third appearance of Jesus after his resurrection.[6]

The precision of the number of fish in this narrative has long been considered peculiar, and many scholars, throughout history, have argued that 153 has some deeper significance. Jerome, for example, wrote that Oppian's Halieutica listed 153 species of fish, although this could not have been the intended meaning of the Gospel writer because Oppian composed Halieutica after the Gospel text was written, and at any rate never gave a list of fish species that clearly adds up to 153.[7][8] It has also been noted that the Tetragrammaton occurs 153 times in the Book of Genesis.[9]

Augustine of Hippo argued that the significance lay in the fact that 153 is the sum of the first 17 integers (i.e. 153 is the 17th triangular number), with 17 representing the combination of divine grace (the 7 gifts of the Spirit) and law (the Ten Commandments).[10][11] Theologian D. A. Carson discusses this and other interpretations and concludes that "If the Evangelist has some symbolism in mind connected with the number 153, he has hidden it well,"[12] while other scholars note "No symbolic significance for the number of 153 fish in John 21:11 has received widespread support."[13]

Writers claiming a major role for Mary Magdalene have noted that in Greek isopsephy her epithet "η Μαγδαληνή" bears the number 8 + 40 + 1 + 3 + 4 + 1 + 30 + 8 + 50 + 8 = 153, thus, it is suggested, revealing her importance.[14] Similarly, the phrase "τὸ δίκτυον" (the net) used in the passage bears the number 1224 = 8 × 153,[14] as do some other phrases. The significance of this is unclear, given that Koine Greek provides a choice of several noun endings[15] with different isopsephy values.[16] The number 153 has also been related to the vesica piscis, with the claim that Archimedes used 153 as a "shorthand or abbreviation"[14] for the square root of 3 in his On the Measurement of the Circle. However, examination of that work[17] does not find the number 153 used in that way.[4]

Evagrius Ponticus referred to the catch of 153 fish, as well as to the mathematical properties of the number (153 = 100 + 28 + 25, with 100 a square number, 28 a triangular number and 25 a circular number) when describing his 153-chapter work on prayer.[18] Louis de Montfort, in his fifth method of saying the Rosary, connects the catch of 153 fish with the number of Hail Marys said (3 plus 15 sets of 10),[19] while St Paul's School in London was founded in 1512 by John Colet to teach 153 poor men's children, also in reference to the catch.[20]

153 is also:

## References

1. ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 140–141.
2. ^ Gupta, Shayam Sunder. "Curious Properties of 153". Retrieved June 26, 2009.
3. ^ OEIS:A046197
4. ^ a b
5. ^ OEIS:A165330
6. ^ Biblegateway John 21:1–14
7. ^ Grant, Robert McQueen (1999). Early Christians and Animals. Routledge. p. 23. ISBN 0-415-20204-3.
8. ^ Grant, Robert M. (October 1949). "One Hundred Fifty-Three Large Fish (John 21:11)". Harvard Theological Review. 42 (4): 273–275. doi:10.1017/s0017816000024329. JSTOR 1508507.
9. ^
10. ^ Jason Byassee, Praise Seeking Understanding: Reading the Psalms with Augustine, Wm. B. Eerdmans Publishing, 2007, p. 130, ISBN 0-8028-4012-4.
11. ^ John E. Rotelle (ed) and Edmund Hill (tr), The works of Saint Augustine: A Translation for the 21st Century, Part 3, Volume 7 (Sermons: On the Liturgical Seasons), p. 112, ISBN 1-56548-059-7.
12. ^ D. A. Carson, The Gospel According to John (Pillar Commentaries Series), Wm. B. Eerdmans Publishing, 1991, p. 673, ISBN 0-85111-749-X.
13. ^ Leland Ryken, Jim Wilhoit, Tremper Longman, Colin Duriez, Douglas Penney, and Daniel G. Reid, Dictionary of Biblical Imagery (Fish), InterVarsity Press, 1998, p. 290, ISBN 0-8308-1451-5.
14. ^ a b c Margaret Starbird, Magdalene's Lost Legacy: Symbolic Numbers and the Sacred Union in Christianity, Inner Traditions / Bear & Company, 2003, pages 49 and 139, ISBN 1-59143-012-7.
15. ^ J.W. Wenham, The Elements of New Testament Greek, Cambridge University Press, 1965.
16. ^ For example, ἰχθύς (fish) has isopsephy values of 1219, 1069, 1289, 1029, 1224, 1220, 1869, 1229, and 1279 with the different noun endings on p. 124 of Wenham, and a further range of possibilities when the definite article is added.
17. ^ Heath, Thomas Little (1897), The Works of Archimedes, Cambridge University, pp. lxxvii ; 50, retrieved January 30, 2010
18. ^ William Harmless, Desert Christians: An Introduction to the Literature of Early Monasticism, Oxford University Press, 2004, pp. 320–341, ISBN 0-19-516222-6.
19. ^ Montfort.org.uk, "its fruitfulness as shown in the net that St. Peter by order of Our Lord threw into the sea and which though filled with 153 fish did not break."
20. ^ Peter Cunningham, Modern London; or, London as it is, 1851, p. 193.
21. ^ Table of transmission frequencies
22. ^ World-airport-coedes.com
23. ^