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.

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").

E 05 +0.00000000000000000001998

54 Errata

the number of total error=1st irregular prime

Error check

the number of total errors in E101=

05 +0.00000000000000000000002

06 +0.00000000000000000000001

07 +0.00000000000000000000000

08 -0.00000000000000000001526

11 -0.00000000000000000000101

16 -0.00000000000000000000001

21 -0.00000000000000000000001

24 -0.00000000000000000000001

67=59+8

(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)

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 ----

(157,62) (157,110)

(37,32) (691,12)

the number of total errors=8th irregular prime 157

You can find many irregular prime numbers somewhere odd.

Reg: 2005, Jul 9

Title: Euler, irregular primes, errors

Method:webpage

Content:

https://math0.pm.tokushima-u.ac.jp/~hiroki/major/euler1-e.html

EP: 37

Index