This cyclotomic number seems to be related to Euler's phi (totient) function ?
Pfgw64 Phi(10^4): 4000
Pari/gp (16:59) gp > eulerphi(10^4) %1 = 4000

What is the role of the first parameter in pfgw64's Phi ?

It provides us the left and the right string of digits around the central digit 1 in the expansion.
Pfgw64 -f0 -od -q"(137*10^11+731*10^7)*(10^3-1)/999" 13707310000000
Pfgw64 -f0 -od -q"(137*10^15+731*10^8)*(10^6-1)/999" 137137073173100000000
Notice the zero between the 137 and 731 string where the middle digit '1' of the Phi function will be added to.

General formula is (137*10^K+731*10^L)*(10^A-1)/999
(10^A-1)/999 is the repeater; with A multiple of 3 (137 and 731 are each three digits long).
L is the number of trailing zeros you want implemented (The last zero will change to a '1' when the right digit '1' of the Phi function is added).
K = L + A + 1

The largest one in the list dating from [ November 14, 1999 ] is
10^35352+2049402*10^17673+1 and is 35353 digits long !
The number of digits, 35353, is a nice undulating palindromic prime (no accident).
I believe it is the largest known prime that is NOT of the form, a*b^n +/- 1.
The estimated search time was 121 days for a Pentium/400 equivalent.
Actual time was about 1/3 of this. It was found by my wife's P/166 computer
(running in the background), the slowest computer that was being used.

The site "The largest known primes"
[http://www.utm.edu/research/primes/ftp/short.txt]
kept monthly updates also about palindromic primes.
The third largest one appearing in the list is
10^19390+4300034*10^9692+1 and is 19391 digits long !
Note that the length of this giant is palindromic too (no accident!).
The second largest one appearing in the list is
10^30802 + 1110111*10^15398 + 1 and is 30803 digits long !
It consists of only 1's and 0's (definitely an accident).
The estimated search time was 233 days of Pentium/200 equivalent.
It was found after only about 12 computer-days or 5% of the estimate.
"With this kind of luck maybe I should start playing the lottery" the author says
who is none other than Harvey Dubner and who already published various articles
about palindromic primes in the Journal of Recreational Mathematics.

[ January 29, 2000 ]
Not afraid to search for larger palprimes himself Warut Roonguthai
used PrimeForm as a tool and this soon turned out to be successful,
though he didn't become record-holder yet...

I've just found the palindromic prime 10^11840+42924*10^5918+1 with
PrimeForm. To search, I used the expression:
(k*1000+100*(k/10)-990*(k/100)-99*(k/1000))*10^(n-3)+1+100^n
It was arranged in such a way that the values of k and n could be seen in
the log file. The formula k*1000+100*(k/10)-990*(k/100)-99*(k/1000)
was used to generate the palindromic middle term from k; k could be any
integer from 1 to 9999.  If the decimal expansion of k is abcd, then the
formula will produce the number abcdcba.  Note that with PrimeForm, the
value of (x/y) is the greatest integer that is equal to or less than the
fraction x/y, i.e. (x/y) = [x/y].