Prime divisors of the Bernoulli and Euler numbers

Samuel S. Wagstaff, Jr.^{∗} Center for Education and Research in Information Assurance and Security

and Department of Computer Sciences, Purdue University West Lafayette, IN 47907-1398 USA

Abstract

We have completely factored the numerators N_{2k} of the Bernoulli numbers for all 2k ≤ 152 and the Euler numbers E_{2k} for all 2k ≤ 88, using the even index notation. We studied the results seeking new theorems about the prime factors of these numbers. We rediscovered two nearly-forgotten congruences for the Euler numbers.

1 Factoring the Bernoulli and Euler numbers

The Bernoulli numbers B_{n} may be defined by the generating function

\frac{t}{e^{t} − 1} = ∑_{n=0}^{∞} B_{n} \frac{t^{n}}{n!} .

The B_{n} are all rational numbers, B_{2k+1} = 0 for all k ≥ 1, and the non-zero B_{n} alternate in sign. The first few non-zero ones are: B_{0} = 1, B_{1} = −1/2, B_{2} = 1/6, B_{4} = −1/30, B_{6} = 1/42, B_{8} = −1/30, B_{10} = 5/66, B_{12} = −691/2730, B_{14} = 7/6, B_{16} = −3617/510, B_{18} = 43867/798. B_{20} is the first one with a composite numerator: 174611 = 283 · 617.

Write B_{n} as N_{n}/D_{n} with D_{n} > 0 and gcd(N_{n}, D_{n}) = 1. It is easy to describe the denominators:

∗ This work was supported in part by the Purdue University CERIAS and by the Lilly Endowment, Inc.

Theorem 1 (von Staudt-Clausen [34, 9] 1840) If n > 0, then

D_{n} = ∏_{{p prime \atop p−1|n}} p, and B_{n} + ∑_{{p prime \atop p−1|n}} \frac{1}{p} is an integer.

If a prime p divides some numerator N_{n}, then it divides every p − 1-st numerator after that:

Theorem 2 (Kummer [19] 1851) If n ≥ 1, p is a prime ≥ 5 and p−1 |̸2n, then

\frac{B_{2n+(p−1)}}{2n + (p − 1)} ≡ \frac{B_{2n}}{2n} mod p.

Another useful fact about the prime factors of N_{n} is this:

Theorem 3 (J. C. Adams [1] 1878) If p is prime, n ≥ 1, p − 1 |̸2n and p^{e}|2n for some e ≥ 1, then p^{e}|N_{2n}.

Slavutskii [27] attributes both Kummer’s congruence and Adams’ theorem to two obscure pamphlets [35] of von Staudt. See also [28]. The Bernoulli numbers and the prime factors of their numerators have been of fundamental importance in the study of cyclotomic fields since the time of Kummer. For example, see Iwasawa [16] and Ribenboim [25]. Before Wiles proved Fermat’s Last Theorem, these numbers provided an important avenue of attack on that problem.

M. Ohm [22] made the first attempt to factor Bernoulli numerators in 1840. In unpublished work, J. Bertrand, J. L. Selfridge, M. C. Wunderlich, and others, factored more Bernoulli numerators. In 1978, we [36] published the factorizations through N_{60}, but there was a typo in the very last factor. Now we have factored N_{2k} for all 2k ≤ 152 and for many larger 2k ≤ 300. See Adams [1] for the unfactored N_{2k} and the D_{2k}. See Knuth and Buckholtz [18] for a simple method of computing these numbers. We used their method to compute the numbers. We publish the factors here to aid the study of cyclotomic fields.

Some other works which consider prime factors of Bernoulli numbers, mostly with large subscripts (far beyond the range of this paper), and which extend the work of [36], include [6, 4, 5] and pages 116ff of [10].

Five tables, placed at the end of this paper to preserve continuity, summarize our efforts over many years to factor the Bernoulli numerators and the Euler numbers. The complete results are available at the web address:

http://www.cerias.purdue.edu/homes/ssw/bernoulli/index.html.

In Table 1, we give the complete factorization of N_{2k} for 60 ≤ 2k ≤ 132. In the tables, Pxx and Cxx denote prime and composite numbers with xx digits, respectively. To keep the paper short, Tables 2 and 3 show only the large (> 11 digits) prime factors. We assume that anyone using the tables can compute the numerators and discover the small factors easily. Several modern computer algebra systems, such as Maple and Mathematica, have Bernoulli and Euler numbers and polynomials as built-in functions. If a numerator is omitted, then we know no large prime factor of it. But the numerator is not omitted if the final known factor is prime. Thus the line “144 P135” in Table 2 means that N_{144} is the product of one or more small primes (in fact, 6500309593) times a 135-digit prime, not that N_{144} is prime.

The Euler numbers E_{n} may be defined by the generating function

\frac{2e^{t/2}}{e^{t} + 1} = ∑_{n=0}^{∞} \frac{E_{n} · t^{n}}{2^{n} · n!} = ∑_{n=0}^{∞} \frac{E_{n}}{n!} (\frac{t}{2})^{n}

or by the formula

sec x = ∑_{n=0}^{∞}(−1)^{n}E_{2n} \frac{x^{2n}}{(2n)!}.

The Euler numbers with odd subscripts vanish: E_{2k+1} = 0 for all k ≥ 0. The non-zero Euler numbers are odd integers which alternate in sign. The first few non-zero Euler numbers are: E_{0} = 1, E_{2} = −1, E_{4} = 5, E_{6} = −61, E_{8} = 1385, E_{10} = −50521, E_{12} = 2702765.

Since the Euler numbers are all integers, there is no analogue for them of the von Staudt-Clausen Theorem. Kummer’s Theorem has an analogue for E_{2n}, also proved by Kummer. We state it as Theorem 4 below. Our search for an analogue to J. C. Adams’ Theorem led to the work in the next section.

The prime factors of the Euler numbers determine the structure of certain cyclotomic fields. See Ernvall and Metsänkylä [12], for example.

Most of the above remarks about factoring Bernoulli numbers apply equally to Euler numbers. We [36] published the factorizations through E_{42} in 1978. Now we have factored E_{2k} for all 2k ≤ 88 and for some larger 2k ≤ 200.

In Table 4, we give the complete factorization (if known) of E_{2k} for 40 ≤ 2k ≤ 112. To save space, Table 5 shows only the large (> 10^{11}) prime factors. We assume that anyone using the tables can compute the Euler numbers and discover the small factors easily. If an Euler number is omitted, then we know no large prime factor of it.

We found most of the factors in the five tables by trial division and the elliptic curve method [20]. The largest two of these factors found by the elliptic curve method were the P42 of E_{150} and the P40 of N_{206}. A few large composite cofactors were finished by the quadratic sieve factoring algorithm [23], including the C114 = P37·P77 of N_{206} and the C112 = P44·P69 of E_{116}. Large primes in these tables were proved prime by the methods of the Cunningham Project [3], including the elliptic curve prime proving method [2] for the large primes. The two largest prime divisors of Bernoulli numerators known to us are the P359 factor of N_{292} and the P332 divisor of N_{298}. The largest known prime divisor of an Euler number is the P278 of E_{194}. No doubt one could easily find larger prime divisors of the Bernoulli and Euler numbers by extending the tables a little. The first incomplete factorizations in the tables are the C123 of N_{154} and the C119 of E_{90}. The elliptic curve method, using several hundred curves with a first phase limit 2 · 10^{6}, has been tried on these numbers and on all the other composites in the tables.

2 Congruences for the Euler numbers

In this section we prove Kummer’s Theorem for Euler numbers and two little-known congruences for Euler numbers which we rediscovered by examining (the full version of) Tables 4 and 5 in search of an analogue for J. C. Adams’ Theorem. We also make some historical remarks about these theorems.

The Euler polynomials may be defined by the generating function

\frac{2e^{xt}}{e^{t} + 1} = ∑_{n=0}^{∞} E_{n}(x)\frac{t^{n}}{n!} .

It is easy to see that E_{n} = 2^{n}E_{n}(1/2), for n ≥ 0, and that E′_{n}(x) = nE_{n−1}(x), for n > 0. These two facts lead easily to the Taylor expansion of E_{n}(x) about x = 1/2:

E_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}\frac{E_{k}}{2^{k}} (x − \frac{1}{2})^{n−k} ,  (1)

which holds for all nonnegative integers n and all real x, and which was proved by Raabe [24] in 1851.

Euler, on page 499 of [14], introduced Euler polynomials to evaluate the alternating sum

A_{n}(m) = ∑_{k=1}^{m}(−1)^{m−k}k^{n} = m^{n} − (m − 1)^{n} + ⋯ + (−1)^{m−1}1^{n},

where m and n are nonnegative integers. The identity E_{n}(x+1)+E_{n}(x) = 2x^{n} follows easily from the definition of Euler polynomials. Alternately adding and subtracting this identity with x = m − 1, x = m − 2, …, x = 1, gives the formula

A_{n}(m) = \frac{1}{2}(E_{n}(m + 1) − (−1)^{m}E_{n}(1))  (2)

for integers m, n ≥ 0. In the same way, one can prove that

C_{n}(b, m) =^{def} ∑_{k=1}^{m}(−1)^{m−k}(k + b − 1)^{n} = \frac{1}{2}(E_{n}(b + m) − (−1)^{m}E_{n}(b)) (3)

for any real b and integers m, n ≥ 0. Setting x = 0 in E_{n}(x + 1) + E_{n}(x) = 2x^{n} shows that E_{n}(1) = −E_{n}(0).

Lemma 1 If n is an even positive integer, then E_{n}(0) = E_{n}(1) = 0.

Proof: Substituting x = 0 and x = 1 in (1) and using the fact that E_{2j+1} = 0, one finds that

E_{n}(0) = 2^{−n}(−1)^{n} ∑_{k=0}^{n}(−1)^{k}\binom{n}{k}E_{k} = 2^{−n} ∑_{k=0}^{n} \binom{n}{k}E_{k} = E_{n}(1).

But we just saw that E_{n}(1) = −E_{n}(0), so E_{n}(1) = E_{n}(0) = 0.

Proposition 1 If p > 0 is odd and n > 0 is even, then

A_{n} (\frac{p − 1}{2}) = 2^{−n−1} ∑_{k=0}^{n} \binom{n}{k}E_{k}p^{n−k}.

Proof: Let x = (p + 1)/2 in (1). One gets

E_{n} (\frac{p + 1}{2}) = ∑_{k=0}^{n} \binom{n}{k}\frac{E_{k}}{2^{k}} (\frac{p}{2})^{n−k} = 2^{−n} ∑_{k=0}^{n} \binom{n}{k}E_{k}p^{n−k}. (4)

Let m = (p − 1)/2 in (2). Thus, m + 1 = (p + 1)/2 and

A_{n} (\frac{p − 1}{2}) = \frac{1}{2} (E_{n} (\frac{p + 1}{2}) − (−1)^{(p−1)/2}E_{n}(1)) .

The proposition now follows from (4) and Lemma 1.

Wells Johnson [17] began with a formula analogous to the one in Proposition 1 and gave p-adic proofs of many facts about Bernoulli numbers, including Theorems 1, 2 and 3. We will use similar methods to prove facts about Euler numbers.

Let e_{p} denote the exponential p-adic valuation on the integers or rational numbers. Thus e_{p}(n) = r means p^{r}||n. We will need Johnson’s lemma, which follows easily from the well-known fact that (p − 1)e_{p}(j!) = j − ∑_{i≥0} d_{i}, where j = ∑_{i≥0} d_{i}p^{i} and 0 ≤ d_{i} < p.

Lemma 2 (Johnson [17] 1975) If p is prime and j ≥ 1, then

e_{p} (\frac{p^{j}}{j!} ) > \frac{p − 2}{p − 1}j.

We begin with the analogue of Kummer’s Theorem mentioned above:

Theorem 4 (Kummer [19] 1851) If n ≥ 1 and p ≥ 3 is prime, then E_{2n+(p−1)} ≡ E_{2n} mod p.

Proof: Write m = (p − 1)/2. Taken modulo p, the formula of Proposition 1 is

A_{2n}(m) ≡ 2^{−2n−1}E_{2n} mod p.

Therefore,

E_{2n} ≡ 2^{2n+1} ∑_{k=1}^{m}(−1)^{m−k}k^{2n} mod p

and

E_{2n+(p−1)} ≡ 2^{2n+(p−1)+1} ∑_{k=1}^{m}(−1)^{m−k}k^{2n+(p−1)} mod p.

But k^{2n+(p−1)} ≡ k^{2n} mod p for 1 ≤ k < p by Fermat’s Little Theorem, and Kummer’s congruence follows.

Carlitz and Levine [8] have also investigated Kummer’s congruence for Euler numbers.

Here is the analogue of J. C. Adams’ Theorem:

Theorem 5 Let p be an odd prime, n a positive integer and e a nonnegative integer. Suppose (p − 1)p^{e} divides n. Then E_{n} ≡ 0 or 2 mod p^{e+1} according as p ≡ 1 or 3 mod 4.

Proof: Write m = (p − 1)/2. By hypothesis, ϕ(p^{e+1}) = (p − 1)p^{e} divides n. The numbers k between 1 and m are relatively prime to p, so k^{n} ≡ 1 mod p^{e+1} by Euler’s Theorem. Thus,

A_{n}(m) = ∑_{k=1}^{m}(−1)^{m−k}k^{n} ≡ ∑_{k=1}^{m}(−1)^{m−k} mod p^{e+1}.

The sum is 0 if m is even, that is, if p ≡ 1 mod 4, and 1 if m is odd, that is, if p ≡ 3 mod 4. Now 2^{−n} ≡ 1 mod p^{e+1} by Euler’s Theorem, so Proposition 1 gives us

E_{n} + ∑_{k=0}^{n−1} \binom{n}{k}E_{k}p^{n−k} ≡ 0 or 2 mod p^{e+1}

according as p ≡ 1 or 3 mod 4.

To prove the theorem, it suffices to show that every term \binom{n}{k}E_{k}p^{n−k}, for 0 ≤ k ≤ n − 1, is divisible by p^{e+1}. Write j = n − k, so that 1 ≤ j ≤ n. Then

e_{p} (\binom{n}{k}E_{k}p^{n−k}) ≥ e_{p} (\binom{n}{k}p^{n−k}) ≥ e_{p}(n) + e_{p} (\frac{p^{j}}{j!} ) .

By hypothesis, e_{p}(n) ≥ e. By Lemma 2, e_{p}(p^{j}/j!) > j(p − 2)/(p − 1). Now j ≥ 1. The fraction (p − 2)/(p − 1) is minimized (over odd primes p) when p = 3. Thus e_{p}(\binom{n}{k}E_{k}p^{n−k}) > e+1(3−2)/(3−1) or e_{p}(\binom{n}{k}E_{k}p^{n−k}) ≥ e+1, which completes the proof.

Theorem 5 shows, for example, that E_{2k} ≡ 2 mod 3, E_{4k} ≡ 0 mod 5, E_{6k} ≡ 2 mod 7, E_{6k} ≡ 2 mod 9 and E_{10k} ≡ 2 mod 11 for all k > 0.

Carlitz [7] gave a proof very similar to the one above.

Now define

D_{n}(m) = ∑_{k=1}^{m}(−1)^{m−k}(2k −1)^{n} = (2m−1)^{n} −(2m−3)^{n} +⋯+(−1)^{m−1}1^{n}

for integers m ≥ 1, n ≥ 0.

Proposition 2 If m ≥ 1 and n ≥ 0, then D_{n}(m) = 2^{n}C_{n} (\frac{1}{2}, m).

Proof:

2^{n}C_{n}(\frac{1}{2}, m) = 2^{n} ∑_{k=1}^{m}(−1)^{m−k}(k− \frac{1}{2})^{n} = ∑_{k=1}^{m}(−1)^{m−k}(2k−1)^{n} = D_{n}(m).

Proposition 3 If m ≥ 1 and n ≥ 0, then

D_{n}(m) = ∑_{k=0}^{n−1} \binom{n}{k}2^{n−k−1}E_{k}m^{n−k} + \frac{1 − (−1)^{m}}{2} E_{n}.

Proof: Using the previous proposition and Equations (3) and (1), we have

D_{n}(m) = 2^{n}C_{n} (\frac{1}{2}, m) = 2^{n−1} (E_{n} (\frac{1}{2} + m) − (−1)^{m}E_{n} (\frac{1}{2}))

= 2^{n−1} (∑_{k=0}^{n} \binom{n}{k}\frac{E_{k}}{2^{k}} m^{n−k} − (−1)^{m} \frac{E_{n}}{2^{n}} )

= ∑_{k=0}^{n} \binom{n}{k}2^{n−k−1}E_{k}m^{n−k} − (−1)^{m} \frac{E_{n}}{2}

= ∑_{k=0}^{n−1} \binom{n}{k}2^{n−k−1}E_{k}m^{n−k} + \frac{1 − (−1)^{m}}{2} E_{n}.

Lemma 3 Let n ≥ 0, k ≥ 1, a and b be integers with a ≡ b mod 2^{k}. (a) If a is odd, then (2n + 2^{k})(b) ≡ (2n)(a) + 2^{k} mod 2^{k+1}.

(b) If a is even, then (2n + 2^{k})(b) ≡ (2n)(a) mod 2^{k+1}.

Proof: Write b = a + c2^{k} for some integer c. Then

(2n + 2^{k})(b) = (2n + 2^{k})(a + c2^{k}) ≡ (2n)(a) + a2^{k} mod 2^{k+1}.

(a) If a is odd, then a2^{k} ≡ 2^{k} mod 2^{k+1}.

(b) If a is even, then a2^{k} ≡ 0 mod 2^{k+1}.

Theorem 6 For all integers n ≥ 0 and k ≥ 0 we have E_{2n} ≡ E_{2n+2^{k}} + 2^{k} mod 2^{k+1}.

Proof: Let m = 1 in Proposition 3. Then

1 = D_{n}(m) = ∑_{i=0}^{n−1} \binom{n}{i}2^{n−i−1}E_{i} + E_{n}

for n ≥ 0. Replace i by n − j in this formula and find that

1 = E_{n} + ∑_{j=1}^{n} \binom{n}{j}2^{j−1}E_{n−j}

for n ≥ 0. Replace n first by 2n and then again by 2n + 2^{k} to get

E_{2n} + ∑_{j=1}^{2n} \binom{2n}{j}2^{j−1}E_{2n−j} = E_{2n+2^{k}} + ∑_{j=1}^{2n+2^{k}} \binom{2n + 2^{k}}{j}2^{j−1}E_{2n+2^{k}−j},

since both sides equal 1. We can rewrite this as

E_{2n} = E_{2n+2^{k}} + ∑_{j=1}^{2n+2^{k}} 2^{j−1} (\binom{2n + 2^{k}}{j}E_{2n+2^{k}−j} − \binom{2n}{j}E_{2n−j}) (5)

because \binom{2n}{j} = 0 when j > 2n. We may ignore the terms with odd j in (5) because E_{2i+1} = 0 for all i ≥ 0. We will show that each term with even j in the sum in (5) is divisible by 2^{k+1}, except the term with j = 2, which we will show is ≡ 2^{k} mod 2^{k+1}.

We now prove the theorem by induction on k. For k = 0 it says E_{2n} ≡ E_{2n+1} + 1 mod 2. This is true because E_{2n+1} = 0 and E_{2n} is odd.

Now let k ≥ 1 and assume that E_{2n} ≡ E_{2n+2^{k−1}} + 2^{k−1} mod 2^{k}. If k ≥ 2, then also by induction E_{2n+2^{k−1}} ≡ E_{2n+2^{k}} + 2^{k−1} mod 2^{k}, so that

E_{2n} ≡ E_{2n+2^{k−1}} mod 2^{k}.  (6)

In fact, (6) holds also when k = 1, since every E_{2n} is odd and E_{2n+1} = 0. The general term in the sum in (5) is

\frac{2^{j−1}}{j!} \{(2n + 2^{k})(2n + 2^{k} − 1) ⋯ (2n + 2^{k} − j + 1)E_{2n+2^{k}−j}

− (2n)(2n − 1) ⋯ (2n − j + 1)E_{2n−j}}.  (7)

By Lemma 2, 2^{j−1}/j! is a 2-integer, and it equals 1 when j = 2. Also, 2n + 2^{k} − i ≡ 2n − i mod 2^{k} for each i. With (6) we have for each even j

(2n + 2^{k} − 1)(2n + 2^{k} − 3) ⋯ (2n + 2^{k} − j + 1)E_{2n+2^{k}−j}
≡ (2n − 1)(2n − 3) ⋯ (2n − j + 1)E_{2n−j} mod 2^{k}.

Each side of this congruence is an odd number. We now multiply both sides by the even factors in (7). Multiply the congruence by the congruent even numbers 2n + 2^{k} − 2i, 2n − 2i, one on each side, for each i, and use Lemma 3. When j = 2, there is just one even factor on each side, we use Lemma 3(a) once, and the number in (7) is ≡ 2^{k} mod 2^{k+1}. When j > 2, there is more than one even factor on each side, we use Lemma 3(a) once, Lemma 3(b) at least once, and the general term in (7) is divisible by 2^{k+1}. This proves the theorem.

Corollary 1 The set {E_{0}, E_{2}, …, E_{2^{k}−2}} forms a reduced set of residues modulo 2^{k} for k ≥ 1.

Proof: Use induction. For k = 1, {E_{0}} = {1} is an RSR modulo 2^{1}. Assume true for k and prove for k+1. By the theorem, E_{2n+2^{k}} ≡ E_{2n}+2^{k} mod 2^{k+1} for n = 0, 1, …, 2^{k−1} − 1. Therefore the statement holds for k + 1.

Theorem 5 and Corollary 1 were stated without proof by Sylvester [31, 33, 30, 32] in 1861. A few years later, Stern [29] gave brief sketches of proofs of these two results and of Theorem 6. In 1910, Frobenius [15] amplified Stern’s sketches of these proofs. Ernvall [13] in 1979 said he couldn’t understand Frobenius’ outline of the proofs and gave his own proofs using the umbral calculus. The case e = 0 of Theorem 5 was proved by Ely [11] and mentioned by Nielsen [21]. These works of Sylvester, Stern and Ely are noted by Saalschütz [26]. Proposition 2 is in Nielsen [21]. Our proofs of Theorems 4, 5 and 6 have the p-adic flavor of proofs of similar statements for the Bernoulli numbers in Johnson [17].

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[35] K. G. C. von Staudt. De numeris Bernoullianis, I, II. Junge, Erlangen, 1845.

[36] S. S. Wagstaff, Jr. The irregular primes to 125000. Math. Comp., 32(142):583–591, 1978.

Table 1: Bernoulli numerators |N_{n}| factored

n Prime factorization of |N_{n}|
60 2003·5549927·109317926249509865753025015237911
62 31·157·266689·329447317·28765594733083851481
64 1226592271·87057315354522179184989699791727
66 11·839·159562251828620181390358590156239282938769
68 17·37·101·123143·1822329343·5525473366510930028227481
70 5·7·688531·20210499584198062453·3090850068576441179447
72 3112655297839·1872341908760688976794226499636304357567811
74 37·923038305114085622008920911661422572613197507651
76 19·58231·22284285930116236430122855560372707885169924709
78 13·787388008575397·33364652939596337·1214698595111676682009391
80 631·10589·5009593·141795949·P39
82 41·4003·38189·P51
84 233·271·68767·167304204004064919523·P37
86 43·541·21563·P55
88 11·307·2682679·P60
90 5·587·1758317910439·P57
92 23·587·108023·P63
94 47·467·1499·2459153·4217126617741589575995641·P34
96 7823741903·4155593423131·10017952436526113·
·96454277809515481·P25
98 7·7·2857·3221·1671211·9215789693276607167·P43
100 263·379·28717943·65677171692755556482181133·P45
102 17·59·827·17833331·86023144558386407·
·299116358909830276447443337·P28
104 13·37·776253902057299·6644689804135385589700423·P45
106 53·3967·37217·77272435237709·P65
108 656884664663·23657486502844933·P69
110 5·157·76493·150235116317549231·36944818874116823428357691·P44
112 7·887569·8065483·P86
114 19·P97
116 29·7559·7438099·6795944986967·P77
118 59·P100
120 6495690221·8070196213·P93
122 61·1545314586433142560447·1545923474257037240728199709913·P54
124 31·67·74747·162263·14066893·8262971607841·
·3498285428145163·16743250272239551·P45
126 103·409·216363744721·P102
128 35089·5953097·12349588663·13349390911530343·
·6996505560116602097773394576621473·P46
130 5·13·149·463·2264267·3581984682522167·P92
132 11·804889·10462099·P112

Table 2: Large prime factors of Bernoulli numerators |N_{n}|

n Large prime factors of |N_{n}|
134 338420464438865099·6005440277888093849051345046242759·P65
136 29835096585483934621·P98
138 554744941981·756906736720877·

·9959596661942153266426403135574603847379·P48

140 44124706530665069·49919098955213994432243162077·P68
142 111781954908479484383981·P105
144 P135
146 22639970526343·6726159702783854797·

·37996324998547740539691528067877·

·1754821172656266926966923716442469·P34

148 4975417507662031677157·

·1248863436460860523032749·P84

150 5810708205829·21664796739499531040947·

·2409795082015672566733218756037·P72

152 3720341037827029338655181363717044961·

·37835716074058426890725596550304118196498159·P52

154 384785986561·C123
156 167604149935534865064907·

·94884267483295622200143616179947·P101

160 40094692599177383·12830086712891890983430059948563·

·1744826505423362390046833266050403703791289·P62

164 104386532651·2903061743891·9898920431428993·C117
166 311318618909·37074748512889·

·60519068332988964084651891032717·

·117092287618059239620235259605532189619·P52

168 19254163575306510187·10094494587919631151637·C128
170 751612064207·P154
172 P174
174 6659961564676431900928667503·C137
176 333026571343·110783038328477·124813394943812621·C138
178 129180506448277·1823634234826012967·39326836920802601519·P129
180 249829228470043·2076252436787489535833·

·4241477436592626145879·P127

182 73107144475261423·311089841618633327·3627027615648746666477·

·2122174114227419648093461601·

·8327616545832330042958707170640293981592673849·P59

184 21983088204089362967·P169
186 922966808867·9161904079472101·C156
190 60860762760882373·174262092707971020104538709609·C152
196 58273617156601282072242637946609·C173
198 723357738211·P201
200 5370056528687·C204

Table 3: Large prime factors of Bernoulli numerators |N_{n}|

n Large prime factors of |N_{n}|
202 85704723183916799·C173
204 9131578873975602379·P207
206 4134128959054219·28391723373218209·408428439912252710783201·

·4794779427824009051318510739603796493· ·3705636735000917624663544925511551624891·P77

216 P239
218 4986305046278328485613904846831·C175
220 792913356669011·C224
222 270574469649607096339·C229
224 67653517083823116668886121·13568731058606497266528850207·P187
226 226941007255811687·C229
230 9487561145259955585249403·C234
232 2483032145171·259051164055671909270473820520219·P225
234 48237362885215689907·C222
236 504680422913·14656891523109995294576720509429987·C219
238 30079831621249·C258
240 26230095767160160157·C260
242 49675522089194103641917241·C236
246 1015348391695196501·C267
248 115134703427104257294711265272763·C240
252 2028290804799829·650932177698080567099·

·130625066385309173899099708579· ·754223542032486571885216433401349· ·1000993741524774643539942570884595839·C171

258 1236523928730271·C292
262 63379712903619825709847·P278
266 167825382335090242001·1727816222222026922465407·

·1571264305785183471309381325703·C216

270 2539833907837164114167·P306
274 21804608848811·201500345265433·31628480989746829·

·3277838401217446489·25729084799117836987901· ·892008372912807309877541·C222

276 116773511307223·9280481761112414180447102368597·C293
280 136100780239·C338
282 4525048629470223385658435650031·C311
284 792213846555737·C331
286 8812943587829·16865476527940273·

·34000751682694166738635652417·C285

288 1259554461969878619108227·C328
292 P359
296 146409753143342542769·C351
298 371472263795653589766634977803·P332
300 7985787872578979·C352

Table 4: Euler numbers |E_{n}| factored

n Prime factorization of |E_{n}|
40 5·5·41·763601·52778129·359513962188687126618793
42 137·5563·13599529127564174819549339030619651971
44 5·587·32027·9728167327·36408069989737·238716161191111
46 19·285528427091·1229030085617829967076190070873124909
48 5·13·17·5516994249383296071214195242422482492286460673697
50 5639·1508047·10546435076057211497·67494515552598479622918721
52 5·31·53·1601·2144617·537569557577904730817·P24
54 43·2749·3886651·78383747632327·P36
56 5·29·5303·7256152441·52327916441·2551319957161·P26
58 1459879476771247347961031445001033·P34
60 5·5·13·47·61·6821509·14922423647156041·P42
62 101·6863·418739·1042901·P56
64 5·17·19·25349·85297·P65
66 61·105075119·508679461·155312172341·P51
68 5·2039·66041·29487071944189·15138431327918641·P45
70 353·2586437056036336027701234101159·P54
72 5·13·37·73·2341·4014623·24259423·30601587075439337·P51
74 193·34629826749613·4207222848740394629·
·22060457167870794468746201·P34
76 5·145007·3460859370585503071·581662827280863723239564386159·P43
78 2740019561103910291228417123054994825316979387·P55
80 5·5·17·41·7701306020743·3572363603188902175396213·P62
82 19·31·4395659·P98
84 5·13·29·4397·739762335239015186706527735192795520726707·P62
86 311·390751·46053168570671·P92
88 5·89·1019·588528876550967927·16292380848703930709213·P72
90 307·C119
92 5·67·7096363493·7308346963823·120476813565517·P85
94 53089·20609829625906839913745698187·
·180986288780569828566819992453·P66
96 5·13·17·43·79·97·835823·2233081·1951860271597317997069749059·
·9416370608392625586845089085196635167·P47
98 71·376003429·5160267661·4363907262506552373343·P94
100 5·5·5·19·101·C134
102 8647·C139
104 5·53·761·2477·P138
106 47·4858416191·98985829942673·
·1150887066548393492521971151372616707·P88
108 5·13·37·109·1462621·8445961·4675063901·C125
110 509053·116904299·134912677·748079839770433·P120
112 5·17·29·31·113·8185757·617575481323·1522046069820268709·
·265053146030428876430329·P94

Table 5: Large prime factors of Euler numbers |E_{n}|

n Large prime factors of |E_{n}|
114 5290253211544727·22557103319451713·
·2565948669867461313318215567·

·118972684453835135392634192556273454718187595705343·P52

116 1098948437923935829829·17698520871521406115634951924463689·
·11661906593316353058846911847709511061777523·P69
118 86611938909696635972683149781·C142
124 545893110893363273374339·C137
126 44305294819613·167237174851562092201·P128
128 91486803609919·33397018471037747·
·38280927951817207·1823694188853227904949904627·
·252181896718842913832793991507441358249·P64
134 321639994822891·214074317717282326017498018953·P148
136 P200
138 12254459673349·34356165690119899·P157
140 P199
142 2978734769·8557612247·P197
144 978576085558923501179·170513218370189155958048891371·P149
146 P213
148 238661068231279·C202
150 13621373428254587·
·111381973999260228282238167431335585433059·C168
152 18051556174129735359181·3957666449530267510589053·
·438321334095183824658294709367·C149
154 139668927262709710013·C210
156 227071134239·P198
158 5519160811451003·C220
162 174175655449·C242
164 63488848774356502730543060633·C209
166 50150236900098278077·C214
168 86771436435012390277·C230
170 70727223023077·1034326231547973051559·P239
172 743155422133·2840083403239·C243
180 6923483330327017·C269
184 2804389579706797633·C284
186 22658461432253·54342802734882461·
·1086110887390889008410968159777·C229
190 559570609330768709·6386014734599369410586902768943·C265
192 1469840300183·6895766514961118059·
·1269672106384218692615790692911·C249
194 28024555486506389·2436437750204310804841·P278
198 2507798651531·49639305210453901009432031·C277
200 16640782677056849·C306