German Wirtschafts-IdNr

What it covers

Returns a valid German Wirtschafts-IdNr (business identification number): a `DE`-prefixed nine-digit core with a correct ISO 7064 MOD 11,10 check digit plus a five-digit distinguishing suffix. Use `suffix` to set it. Set `invalid=true` for a number with a deliberately wrong check digit.

How it’s calculated

The nine-digit core carries a single ISO/IEC 7064 MOD 11,10 check digit computed over the eight payload digits by an iterative recurrence; the DE prefix and five-digit distinguishing suffix carry no checksum.

Start product = 10.
d=1: sum=(1+10)%10=1 (not 0); product=(1*2)%11=2.
d=3: sum=(3+2)%10=5; product=(5*2)%11=10.
d=5: sum=(5+10)%10=5; product=(5*2)%11=10.
d=7: sum=(7+10)%10=7; product=(7*2)%11=14%11=3.
d=9: sum=(9+3)%10=2; product=(2*2)%11=4.
d=2: sum=(2+4)%10=6; product=(6*2)%11=12%11=1.
d=4: sum=(4+1)%10=5; product=(5*2)%11=10.
d=6: sum=(6+10)%10=6; product=(6*2)%11=12%11=1.
Check digit = (11 - 1) % 10 = 10 % 10 = 0.

Edge cases

• The check digit is fully determined by the eight payload digits, so the MOD 11,10 recurrence always produces exactly one decimal digit (0-9) — there is no unrepresentable residue and no redraw case.
• The distinguishing five-digit suffix (default 00001) has no checksum of its own; it is an integer in 1..99999 rendered as exactly five zero-padded digits.
• invalid:true corrupts only the ninth (check) digit: the correct digit is offset by a random 1..9 mod 10, guaranteeing it never lands back on the valid value.

Gotchas

• The checksum covers only the nine-digit numeric core — not the literal DE prefix nor the -NNNNN suffix. Feeding DE or the suffix into the recurrence gives wrong results.
• This is the recursive ISO 7064 MOD 11,10 system (iterative product folding), NOT a weighted-sum mod-11 scheme; do not confuse it with the plain mod11.ts weighting.
• The 0->10 remap inside the loop (when `sum` computes to 0) is essential — omitting it changes the result.