1. The biblical question comes first

Our starting question is theological: how does Scripture describe the relationship between the Father’s purpose, the Son’s work and the community gathered through that work? Across the selected passages, unity is more than proximity. It involves belonging to Christ, participating in His life and being held within the Father’s saving purpose. We therefore begin with the texts before asking whether a mathematical invariant can provide a useful symbolic expression of this relationship.

These passages have complementary roles. Ephesians and John 17 establish the principal themes of gathering and unity. John 6 and John 21 connect those themes to preservation and to the number 153. Their differences matter: cosmic gathering, the unity of believers, the promise of resurrection and a resurrection appearance are related themes, but they are not interchangeable accounts.

2. Ephesians and the gathering of all things in Christ

Ephesians 1:9–10 belongs to the opening blessing of the letter, in which God’s purpose is expressed through election, redemption and the revelation of His will. Verse 9 presents that purpose as a disclosed mystery; verse 10 describes its fulfillment through the gathering of things in heaven and on earth in Christ. Christ is the center of this gathering, rather than merely another member of it.

This supplies the widest horizon for our interpretation: multiplicity is brought into a common relation to Christ according to God’s purpose. The scope of “all things” exceeds the community of believers. Accordingly, we use Ephesians to ground the theme of sovereign gathering, while John 17 more specifically grounds the communal dimension of our formulation. We do not treat the arithmetic as settling debates about the ultimate salvation of every individual.

3. John and the unity for which Jesus prays

John 17:21–23 follows Jesus’ prayer for those who will believe through the disciples’ testimony. The unity He seeks is grounded in the relationship of the Father and Son and expressed through believers’ participation in that relationship. It also has a public purpose: the world is to recognize the Father’s sending of the Son and His love for those who belong to Him.

The people remain many, yet they are called into one shared life. This provides the clearest communal foundation for our proposed reading of the ratio. The unity is relational and Christ-centered; it does not erase personal distinction or turn the Church into the divine being. Together, Ephesians and John 17 offer a coherent theological movement: the Father purposes a gathering in Christ, and the Son prays for the unity of those who believe in Him.

4. The verse identifiers and their sums

In our method, a verse identifier is the sum of its book number, chapter number and verse number, using the conventional ordering of the 66-book Protestant Bible. Ephesians is book 49 and John is book 43. A passage identifier is the sum of the identifiers of its constituent verses. These are values assigned by the method; they are distinct from numbers stated in the biblical text.

The passage sum for Ephesians 1:9–10 is 59 + 60 = 119.

The passage sum for John 17:21–23 is 81 + 82 + 83 = 246.

5. What the Euler–Carmichael ratio measures

Euler’s totient counts the positive integers up to a positive integer n that are relatively prime to it; we denote this count by φ(n). Carmichael’s function gives the least positive exponent that sends every invertible residue class modulo n to 1; we denote it by λ(n). For n ≥ 2, the Euler–Carmichael ratio is defined as R(n) = φ(n)/λ(n).

In group-theoretic terms, φ(n) is the number of elements in the multiplicative group of units modulo n, and λ(n) is its exponent. This group is finite and abelian, so it contains an element whose order equals its exponent. The cyclic subgroup generated by that element therefore has λ(n) elements. The ratio R(n) is the index of such a subgroup: the number of its cosets, or equal-sized parts into which it partitions the group.

Consequently, R(n) is a positive integer, and R(n) = 1 exactly when the unit group is cyclic. It does not count independent cycles, maximal cyclic subgroups or generators. These distinctions keep the symbolic interpretation anchored to the invariant’s actual mathematical meaning.

For readers without a mathematical background

The ratio compares two sizes: the size of a whole collection and the size of the largest cycle that one member can generate by repeated multiplication. Euler’s totient gives the first size; Carmichael’s function gives the second. Dividing the whole collection’s size by the cycle’s size tells us how much larger the collection is than that cycle.

For example, a collection of eight members with a largest cycle of four has ratio 8 ÷ 4 = 2. The cycle reaches half the collection. In this mathematical setting, the collection can be partitioned into two equal-sized groups of four associated with that cycle. This does not mean that there are only two possible cycles.

We can see this with the small number 15. The numbers below 15 that have no common factor with it other than 1 are 1, 2, 4, 7, 8, 11, 13 and 14. There are eight, so Euler’s totient is 8. To follow their multiplication, we keep only the remainder after division by 15. For example, 16 has remainder 1.

Start with 1 and repeatedly multiply by 2, keeping the remainder each time. We obtain the cycle shown below. After four steps we return to 1, having visited four different members: 1, 2, 4 and 8.

No member generates a longer cycle here. Starting from 1, the cycle lengths generated by 1, 2, 4, 7, 8, 11, 13 and 14 are respectively 1, 4, 2, 4, 4, 2, 4 and 2. All these lengths divide 4, and some equal 4. Carmichael’s function is therefore 4. The ratio is 8 ÷ 4 = 2.

The four members in the displayed cycle form one group, {1, 2, 4, 8}. Multiplying each of them by 7 and keeping the remainder gives the other group, {7, 14, 13, 11}. Together these two groups contain all eight members. This is what the technical phrase “index of a cyclic subgroup of maximal order” means in this example.

For our two biblical passage sums, the same comparison operates with larger collections. For 119, there are 96 members and a largest cycle of 48, giving 96 ÷ 48 = 2. For 246, there are 80 members and a largest cycle of 40, giving 80 ÷ 40 = 2. The collections differ in size, yet each is twice as large as its largest cycle.

This example explains the mathematics. Our theological reading is a further interpretive step: the relationship between a whole collection and its internal order offers an analogy for many people gathered into a shared life under divine sovereignty. The arithmetic itself does not identify its members with believers or establish that theological meaning.

Why this ratio matters beyond our example

Understanding whether one cycle can reach the whole collection. In mathematics, the ratio tells us whether repeated multiplication by a suitable number can visit every allowed member. A ratio of 1 means such a multiplier exists. A ratio greater than 1 means no single multiplier can reach the entire collection from 1. For 15, the ratio is 2: even a longest cycle visits only four of the eight members. This makes the ratio a useful summary of the collection’s structure.

Counting cycles in repeated calculations. Suppose a computer repeatedly multiplies by a fixed allowed number and keeps the remainder. If that multiplier generates a longest cycle, the ratio gives the number of separate cycles among all allowed starting values. For 15, multiplication by 2 gives two cycles of four. Knowing this helps us understand how many starting values are needed to explore the whole collection. The qualification matters: multiplication by 4 gives shorter cycles, so the ratio does not count the cycles for every possible multiplier.

Understanding exponent rules used in RSA (Rivest–Shamir–Adleman), a public-key cryptographic system used for encryption and digital signatures. RSA cryptography uses multiplication and powers with remainders to support encryption and digital signatures. Its key equations use Carmichael’s function as a common exponent governing those operations. The RSA specification, RFC 8017, requires the product of the public and private exponents to leave remainder 1 after division by λ(n). The ratio explains how this common exponent compares with Euler’s totient: λ(n) = φ(n)/R(n). It is Carmichael’s function that enters the key condition directly; the ratio describes its relationship to the size of the unit group.

A small arithmetic illustration. For 15, every allowed number raised to the fourth power has remainder 1. Euler’s totient, 8, also supplies an exponent with that property, but Carmichael’s value, 4, is the smallest common one. The ratio 8/4 = 2 records how these two exponent values compare. This is an illustration of the arithmetic underlying exponent conditions, not a usable cryptographic example: real RSA uses much larger numbers and specified encoding schemes. The ratio alone is not a measure of security.

These uses explain why the ratio is worth studying independently of theology. It connects the size of a multiplicative collection with the cycles and exponent rules governing it. Our biblical interpretation is an additional symbolic reading of that mathematical relationship.

6. Different prime structures and the same ratio

The least common multiple of integers is the smallest positive integer divisible by each of them; we denote it by lcm. For our two passage sums, the calculations are:

The common ratio has a transparent arithmetic explanation. The unit-group orders contributed by 7 and 17 are 6 and 16; their greatest common divisor is 2. Since the product of two positive integers divided by their least common multiple equals their greatest common divisor, the ratio for 119 is 2. For 246, the factor 2 contributes a trivial unit group, while 3 and 41 contribute orders 2 and 40. Again, the product divided by the least common multiple is 2.

Thus the finding is not equality of the totients or equality of the exponents. Those values differ. It is equality of their relationship: in both cases, the full unit group has twice the size of a cyclic subgroup of maximal order. This shared relationship makes the comparison mathematically precise without implying that the groups themselves are identical.

7. The proposed theological interpretation

Our Canon associates Euler’s totient with the set-apart remnant and Carmichael’s function with sovereign order. Combining these readings, we propose the following formulation:

Ephesians supplies the Father’s purpose and the centrality of Christ; John 17 supplies the unity of the believing community. The shared ratio of 2 offers an arithmetic correspondence between these related texts. We do not identify the two cosets with the Father and Son, two peoples or two covenants: such identifications would require additional justification. Nor does a larger ratio imply greater holiness. The proposed meaning belongs to the relationship between multiplicity and common order, interpreted in the light of the passages.

8. John 6 and John 21 as the bridge to 153

John 6:39 states the Father’s will that the Son lose nothing of what has been given to Him and raise it at the last day. Here gathering is joined to preservation and resurrection. John 21:11 recounts the landing of 153 large fish and specifically notes that the net remained unbroken. The narrative explicitly supplies both the number and the image of a multitude held together.

In The Lord’s Prayer: A Mathematician’s Creed, chapter 2, especially printed pages 35–37, we read these passages together and interpret 153 as the fulfillment of the Father’s will in His Son, Jesus Christ. The present paper adopts that established interpretation. The gathering of the fish within an unbroken net becomes a symbolic image of those entrusted to the Son being gathered and preserved.

This connection is an intertextual theological reading. John 21:11 does not itself explain 153 by explicitly referring to John 6:39. The relationship arises through the shared Johannine themes of the Father’s gift, the Son’s saving work and preservation. It does not imply that only 153 people are saved.

The numerical distinctions remain clear. John 6:39 has the verse identifier 43 + 6 + 39 = 88. John 21:11 has the identifier 43 + 21 + 11 = 75. Neither identifier is 153. The number 153 enters through the fish count stated in John 21:11; John 6:39 supplies its theological interpretation within our framework.

9. From the passage sum of 365 to the wider investigation

The two passages supply our starting point

We can take the present investigation one step further without introducing another starting number. The identifier sums of Ephesians 1:9–10 and John 17:21–23 are 119 and 246. Adding them gives 365. This combined passage sum arises directly from the principal biblical texts selected for their themes of gathering and unity.

The number 365 has prime factorization 5 × 73. Its Euler totient is therefore 4 × 72 = 288, while its Carmichael value is the least common multiple of 4 and 72, namely 72. Its prime count is also 72. Thus two different constructions—counting primes and finding the common exponent of the unit group—produce the same value.

Consequently, the Euler–Carmichael ratio of the combined passage sum is R(365) = 288/72 = 4. The two individual passage sums each have ratio 2, while their combined sum has ratio 4. This is an exact correspondence for these integers, not a general rule that Euler–Carmichael ratios add when their arguments are added. The concurrence of the prime count and Carmichael value is an additional feature of 365 that merits investigation; we do not assign 72 a new theological meaning here.

A comparison with the previously studied number 198

This passage-based starting point can now be brought into conversation with a wider investigation of the Gospel and the Lord’s Prayer. That work began with three numbers already studied within our Biblical Mathematics framework: 108, 135 and 153. In that framework, 108 represents the threshold of discernment and divine turning toward God; 135 is associated with the compassion and saving purpose of the Father; and 153 represents the fulfillment of the Father’s will in His Son.

The association of 135 with fatherly compassion is illustrated by Psalm 103:13, whose verse identifier is 19 + 103 + 13 = 135. Adding the three signatures gives 396; taking half their sum produces 198. This is how 198 entered the earlier research. The selection of half the sum was an exploratory construction, rather than a calculation prescribed by Scripture.

The sum-of-divisors function, denoted σ(n), adds all positive divisors of n. Comparing 198 with the directly obtained passage sum of 365 gives two exact differences:

In Stephen E. Jones’s The Biblical Meaning of Numbers from One to Forty, 24 is associated with Priesthood and 27 with Ministry of Salvation. These associations invite contextual study of how priestly service and saving ministry relate to gathering, unity and preservation. The arithmetic establishes the differences; it does not itself establish their theological significance. Moreover, these identities compare 365 with an already selected 198: they do not independently derive or uniquely select 198 from the passages.

A direct pathway from 198 to Luke’s Lord’s Prayer

The choice of 198 gains further support from a direct arithmetic pathway to the Lord’s Prayer. Since 198 = 2 × 3² × 11, its Euler totient is φ(198) = 198 × (1 − 1/2) × (1 − 1/3) × (1 − 1/11) = 60. The sum of the positive divisors of 60 is 168.

The aliquot sum is the sum of the proper positive divisors, excluding the number itself; we denote it by s(n). Thus s(n) = σ(n) − n, and s(60) = 168 − 60 = 108. The same intermediate value therefore leads to both 168 and 108.

The connection to 168 is especially concrete. Luke is book 42 in our verse-identification method, so the identifiers of Luke 11:2–4—the passage containing Luke’s account of the Lord’s Prayer—are 55, 56 and 57. Their sum is (42 + 11 + 2) + (42 + 11 + 3) + (42 + 11 + 4) = 168.

This pathway links 198 with the passage identifier of Luke’s prayer and returns to 108, one of the signatures from which 198 was originally constructed. It therefore gives a further mathematical reason to retain 198 in the investigation. Within our interpretive framework, the return to 108 recalls the threshold of discernment and divine turning toward God.

The two outcomes are linked consequences of reaching 60: the aliquot sum follows by subtracting 60 from its divisor sum. They are not independent pieces of evidence, and the pathway does not uniquely select 198. It supports studying the base; it does not prescribe the exponent 10. The theological rationale and arithmetic correspondences for examining the tenth power are described next.

Why the tenth power entered the exploration

Having established the connections of 198 with the Father’s saving purpose and Luke’s account of the Lord’s Prayer, we now examine its tenth power. The number 10 has a relevant theological association: Stephen E. Jones identifies it with Divine Order, Law, relating it to the establishment of divine order through law and judgment. Within our framework, the exponent may therefore symbolize the Father’s saving purpose fulfilled in Jesus Christ operating within divine order and law.

This interpretation gains contextual resonance from Jones’s account of 27 as Ministry of Salvation. He relates 27 to 17 + 10, joining victory with law and describing the ministry of salvation as including the revelation of God’s law. The emphasis here is the fulfillment of the Father’s purpose in Christ; it does not suggest that salvation is earned through law-keeping.

This gives the tenth power a theological rationale for study, while leaving the mathematical operation and its symbolic reading distinct. Jones supplies the association of 10 with divine order and law; applying that association to an exponent is our interpretive proposal. The arithmetic correspondences were recognized before this interpretation of the exponent was developed.

The tenth power of 198 has 23 decimal digits whose sum is 108. Under a specified partition into eight consecutive components—an initial two-digit block followed by seven three-digit blocks—a further component-wise digit summation gives 27. These exact properties invite us to investigate whether the construction offers a coherent symbolic portrait of the Gospel expressed in the Lord’s Prayer. The expansion and calculations follow.

The first digit sums of the eight displayed components are 11, 14, 13, 12, 10, 21, 21 and 6, which total 108. Summing the digits of each of these eight values gives 2, 5, 4, 3, 1, 3, 3 and 6, which total 27. The second total is therefore a component-wise calculation, not the digit sum of 108, which is 9. For arithmetic purposes, the final block 024 is interpreted as 24.

The occurrence of 27 in the prime-count difference between 365 and 198 provides a correspondence with this component-wise total of 27. That equality does not mathematically require the tenth power; the association of 10 with divine order and law supplies a separate theological rationale. The exponent, decimal representation and partition are stated explicitly so that readers can reproduce and assess the construction.

Where the Euler Carmichael ratio enters

Applying the Euler–Carmichael ratio to the eight components gives, in their displayed order, 2, 4, 2, 16, 4, 2, 12 and 4. Their sum is 46. The ratio provides a common way to examine the multiplicative structures of components with different prime factorizations.

The radical of a positive integer is the product of its distinct prime factors; we denote it by rad(n). Since the first component is 92 = 2² × 23, its radical is 46. Thus the component-ratio sum reproduces the radical of the first component:

This is an exact identity under the prescribed partition. The sum is an aggregate of eight ratios, rather than the Euler–Carmichael ratio of the full power. The biblical foundations developed in this paper offer a vocabulary for contemplating that aggregate as multiplicity gathered into unity under divine sovereignty. Ephesians 1:9–10 and John 17:21–23 ground the interpretation; John 6:39 and John 21:11 connect gathering and preservation to our reading of 153.

The order of the argument is therefore important. Scripture supplies the themes and the combined identifier sum of 365. Comparisons with the previously selected 198 then connect the present study to a wider numerical investigation. The tenth-power construction is an application of the proposed interpretation, rather than its biblical foundation. Whether it intentionally encodes the Lord’s Prayer remains an open question.

10. Scope and conclusion

Three claims can now be stated with appropriate precision. The scriptural passages present related themes of gathering, unity and preservation. Their passage sums of 119 and 246 both have an Euler–Carmichael ratio of 2. Within our Canon, that correspondence supports a proposed symbolic reading of the ratio as multiplicity gathered under divine sovereignty.

The ratio of 2 occurs for many other integers. Its presence here therefore cannot establish uniqueness or intentional encoding. Our identifier method also depends on a chosen book order and conventional chapter and verse divisions. Further comparative work can examine whether similar themes and ratios recur under consistently specified selection rules.

Nevertheless, the interpretation has a coherent theological foundation. Ephesians directs our attention to the Father’s purpose in Christ; John 17 to the unity for which the Son prays; John 6 to His promise of preservation and resurrection; and John 21 to the multitude gathered within an unbroken net. In our established reading of 153, these movements meet in the fulfillment of the Father’s saving will in His Son. The ratio offers a mathematical analogy through which to contemplate that unity while preserving the integrity of both the mathematics and the biblical texts.

11. References

The Holy Bible, King James Version. Principal passages: Ephesians 1:9–10; John 17:20–23; John 6:39; John 21:11. Scripture references are cited in the body; wording is mainly paraphrased.

Vanualailai, Jito, Eroni Tomasi, Paulo Vanualailai and Jope Takala. The Lord’s Prayer: A Mathematician’s Creed. Supplied edition, chapter 2, printed pages 35–37.

Jones, Stephen E. The Biblical Meaning of Numbers from One to Forty. Supplied edition, entries for Ten (Divine Order, Law), printed p. 13; Twenty-Four (Priesthood); and Twenty-Seven (Ministry of Salvation), printed pp. 43–44.

Moriarty, K., B. Kaliski, J. Jonsson and A. Rusch. PKCS #1: RSA Cryptography Specifications Version 2.2. RFC 8017, November 2016, Sections 3.1–3.2. RSA key conditions and Carmichael’s function.