## Euler, irregular primes, errors

Japanese

### Leonhard Euler (1707-1783)

I deduce that Euler was the first person who discovered the smallest irregular prime number 37 in zeta values around 1735. Kummer was the first person who discovered it in class numbers of p-cyclotomic fields around 1850. Euler's discoveries seem to have been hidden in errors of the six (or seven) lists of approximated values in E101("Introduction to the Analysis of the Infinite", volume 1). Furthermore the error checks seem to have been hidden in the errors of some values (log pi, cos 1, sin 1 etc.) in E102 (volume 2).
Twenty years after the publishment of E101, he also gave explicit/implicit answers in E343 "Letters to a German princess" and E352 "Remarks on a beautiful relationship between series of powers and reciprocals of powers".
Why did he hide them? It would be very interesting to deliberate the reason. It is really deep and essential. The following contents are hints and answers.

Leonhard Euler is really great. Exponential and Logarithm are really great, too.
Let's challenge to Euler's enigma!

EP: 7 EP: 37 EP: 73
UBASIC Program: E101-0.UB E101-1.UB E101-2.UB E101-3.UB
E101-A.UB E101-B.UB E101-C.UB E102.UB
Important hints and answers were also given in E352 ("Remarks on a beautiful relationship between series of powers and reciprocals of powers") and E343, E344 and E417 ("Letters to a German princess").

## 37=the smallest irregular prime number=12th prime

E101 p.131 p.150 p.151 E212 p.335 p.336 E352 p.87 p.88

zeta(2n)(1-1/2^(2n))

E 05 +0.00000000000000000001998

E: 1998=54*37
54 Errata

the number of total error=1st irregular prime 37

Error check
the number of total errors in E101=54=(1+7+8)+(3+28+7)

approximated values

## Irregular prime numbers and indices

E101 p.153 p.154

E 05 +0.00000000000000000001998

zeta(2n)/2^(2n)

05 +0.00000000000000000000002
06 +0.00000000000000000000001
07 +0.00000000000000000000000

08 -0.00000000000000000001526
11 -0.00000000000000000000101
16 -0.00000000000000000000001
21 -0.00000000000000000000001
24 -0.00000000000000000000001

13=E5+8 +0.00000000000000000000472

472=8*59
67=59+8

101

(37,32=2^2*8)
(59,44=2^2*11)
(67,58=2^1*(13+16))
(101,68=2^1*(13+21))
(103,24=2^0*24)

the number of total errors=7 irregular primes (37, 59, 67, 101, 103, 131, 149)

## 157=the first prime number with the irregular index>1=37th prime

E101 p.237 p.238

the sum of 1/p^(2n)

02 0.452247420041065 +157
04 0.076993139764246 +006
06 0.017070086850637 +002
08 0.004061405366518 -003
10 0.000993603574437 -804
12 0.000246026470035 -002
14 0.000061244396725 ----
16 0.000015282026219 ----
18 0.000003817278702 ----
20 0.000000953961124 -001
22 0.000000238450446 ----
24 0.000000059608184 ----
26 0.000000014901555 ----
28 0.000000003725333 ----
30 0.000000000931326 -003
32 0.000000000232830 ----
34 0.000000000058207 ----
36 0.000000000014551 ----

157

804=37+59+67+101+103+131+149+157=12*67

(157,62) (157,110)
(37,32) (691,12)

the number of total errors=8th irregular prime 157

approximated values and (157,62), (157,110)
You can find many irregular prime numbers somewhere odd.
Reg: 2005, Jul 9
Title: Euler, irregular primes, errors
Method:webpage
Content:
http://hiro2.pm.tokushima-u.ac.jp/~hiroki/major/euler1-e.html
EP: 37 Index