Tax account number

The tax account number is a tax identification number ( English Taxpayer Identification Number , TIN ) and is assigned by the Austrian tax office when a new tax file is created. It is used to clearly identify taxpayers and to make administration efficient. The tax account number is also used as sort criterion referred.

The tax account number is made up of the tax office number and tax number . In everyday language usage , the term tax number is often used as a synonym for the tax account number actually intended.

Since 2012, for example, the term "tax account number (tax office number - tax number)" has been used consistently on the forms provided by the tax office, previously the term "tax number".

General

The tax account number is indicated on all written copies of the tax office (notices, reservations, etc.). The tax account number must always be stated on all receipts (documents, payment coupons, etc.) that are sent to the tax office.

One and the same taxpayer can generally be assigned not just one but also several tax account numbers, each tax account number being intended for a specific purpose. Based on a specific tax account number, the taxpayer can always be clearly identified (by the tax office) to whom this tax account number is assigned.

A tax account number is assigned by the local tax office that is responsible for the taxpayer's place of residence. If the taxpayer changes his place of residence, his tax account number can also change.

Neither the tax account number, the tax office number nor the tax number are listed on official proof of identity . There are also no official cards issued on which the numbers mentioned could be found. On tax assessments, for example, the tax account number is always listed in the upper right corner of the first page.

construction

The tax account number consists of nine digits . It is made up of the tax office number and the tax number together with the check digit . It is structured as follows:

• FA-NNNNNN-P or
• ${\displaystyle (F,A,N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},P)}$ ... as 9- tuple shown

Where:

• FA or and  ... tax office number (always 2-digit, possibly with a leading zero ) ${\displaystyle F}$ ${\displaystyle A}$
• NNNNNN or  … front part (always 6 digits) of the tax number ${\displaystyle N_{1},\dotsc ,N_{6}}$
• P or  … check digit (always 1-digit) as the last digit of the tax number ${\displaystyle P}$

The tax number always has 7 digits and is structured as follows : NNNNNN-P . It consists of the 6-digit front part NNNNNN with the check digit P attached . The check digit P always relates to the tax account number (from the point of view of the calculation), although it is considered to be associated with the tax number in terms of usage and spelling.

Each of the through F , A , N and P , or , , , , , , , and symbolized digit is a digit from the decimal system and can therefore values from the amount accept. ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle N_{1}}$ ${\displaystyle N_{2}}$ ${\displaystyle N_{3}}$ ${\displaystyle N_{4}}$ ${\displaystyle N_{5}}$ ${\displaystyle N_{6}}$ ${\displaystyle P}$ ${\displaystyle \{0,\dotsc ,9\}}$

Note: The hyphen " - " has nothing to do with the spelling, it is only used here to clearly distinguish the individual components from one another and thus to illustrate the structure of the tax account number.

Notation

Different spellings are used for the tax account number: Wherever the best possible legibility is important, especially in paper or document-based processes, separators are usually inserted to make it easier for people to record them as error-free and fatigue-free as possible. For electronic data processing , on the other hand, all separators should preferably be omitted and the nine digits should therefore be written one after the other.

The following notations are common:

• FA-NNN / NNNP  ... easy to read (example: 90-123 / 4567 )
• FA NNN / NNNP  ... easy to read (example: 90 123/4567 )
• FA NNNNNNP  ... poor legibility (example: 90 1234567 )
• FANNNNNNP  ... preferred for electronic data processing (example: 901234567 )

Note: The characters " - " ( hyphen ) and " / " ( slash ) as well as the spaces are only separators for the purpose of better readability and in particular not mathematical operators .

Stock of values

While the tax office number FA is fixed for a certain tax office ( see table ) and the check digit P is determined by a calculation rule ( see below ), the six digits NNNNNN in the front part of the tax number are when a (new) tax account number is generated (by the tax office ) basically freely selectable.

This means that a maximum of 10 ⁶ = 1,000,000 different tax account numbers or tax numbers can be assigned per tax office number. A maximum of 10 ² = 100 different tax office numbers are possible, 40 of which are currently in use. In total there can be 10 ^{2 + 6} = 10 ⁸ = 100,000,000 different tax account numbers.

Tax office numbers

Each tax office is assigned a unique, always 2-digit tax office number FA as follows. The tax office number is always the prefix of every tax account number.

Tax office numbers

FA	Tax office	state	IBAN	"Fictitious" tax account number
03	Vienna 3/6/7/11/15 Schwechat Gerasdorf	Vienna	AT87 0100 0000 0550 4037	03-999 / 9032
04	Vienna 4/5/10	Vienna	AT92 0100 0000 0550 4044	04-999 / 9048
06	Vienna 8/16/17	Vienna	AT26 0100 0000 0550 4068	06-999 / 9068
07	Vienna 9/18/19 Klosterneuburg	Vienna	AT31 0100 0000 0550 4075	07-999 / 9074
08	Vienna 12/13/14 Purkersdorf	Vienna	AT36 0100 0000 0550 4082	08-999 / 9080
09	Vienna 1/23	Vienna	AT62 0100 0000 0550 4099	09-999 / 9096
10	for fees, transfer taxes and gambling	-	AT83 0100 0000 0550 4109	10-999 / 9102
12	Vienna 2/20/21/22	Vienna	AT93 0100 0000 0550 4123	12-999 / 9124
15th	Amstetten Melk Scheibbs	Lower Austria	AT32 0100 0000 0550 4154	15-999 / 9150
16	Baden Mödling	Lower Austria	AT37 0100 0000 0550 4161	16-999 / 9166
18th	Gänserndorf Mistelbach	Lower Austria	AT68 0100 0000 0550 4185	18-999 / 9188
22nd	Hollabrunn Korneuburg Tulln	Lower Austria	AT28 0100 0000 0550 4226	22-999 / 9222
23	Waldviertel	Lower Austria	AT33 0100 0000 0550 4233	23-999 / 9238
29	Lilienfeld St. Pölten	Lower Austria	AT08 0100 0000 0550 4295	29-999 / 9292
33	Neunkirchen Wr. Neustadt	Lower Austria	AT65 0100 0000 0550 4336	33-999 / 9336
38	Bruck Eisenstadt Oberwart	Burgenland, Lower Austria	AT14 0100 0000 0550 4381	38-999 / 9384
41	Braunau Ried Schärding	Upper Austria	AT54 0100 0000 0552 4419	41-999 / 9412
46	Linz	Upper Austria	AT03 0100 0000 0552 4464	46-999 / 9460
51	Church village Perg Steyr	Upper Austria	AT65 0100 0000 0552 4512	51-999 / 9510
52	Free city of Rohrbach Urfahr	Upper Austria	AT91 0100 0000 0552 4529	52-999 / 9526
53	Gmunden Vöcklabruck	Upper Austria	AT96 0100 0000 0552 4536	53-999 / 9532
54	Grieskirchen Wels	Upper Austria	AT04 0100 0000 0552 4543	54-999 / 9548
57	Klagenfurt	Carinthia	AT92 0100 0000 0556 4572	57-999 / 9574
59	St. Veit Wolfsberg	Carinthia	AT26 0100 0000 0556 4596	59-999 / 9596
61	Spittal Villach	Carinthia	AT52 0100 0000 0556 4613	61-999 / 9618
65	Bruck Leoben Mürzzuschlag	Styria	AT73 0100 0000 0553 4650	65-999 / 9650
67	Eastern Styria	Styria	AT07 0100 0000 0553 4674	67-999 / 9672
68	Graz city	Styria	AT12 0100 0000 0553 4681	68-999 / 9688
69	Graz area	Styria	AT38 0100 0000 0553 4698	69-999 / 9694
71	Judenburg Liezen	Styria	AT64 0100 0000 0553 4715	71-999 / 9716
72	Deutschlandsberg Leibnitz Voitsberg	Styria	AT69 0100 0000 0553 4722	72-999 / 9722
81	innsbruck	Tyrol	AT31 0100 0000 0554 4815	81-999 / 9814
82	Kitzbühel Lienz	Tyrol	AT36 0100 0000 0554 4822	82-999 / 9820
83	Kufstein Schwaz	Tyrol	AT62 0100 0000 0554 4839	83-999 / 9836
84	Landeck Reutte	Tyrol	AT67 0100 0000 0554 4846	84-999 / 9842
90	St. Johann Tamsweg Zell am See	Salzburg	AT90 0100 0000 0555 4908	90-999 / 9906
91	Salzburg city	Salzburg	AT95 0100 0000 0555 4915	91-999 / 9912
93	Salzburg country	Salzburg	AT29 0100 0000 0555 4939	93-999 / 9934
97	Bregenz	Vorarlberg	AT37 0100 0000 0557 4971	97-999 / 9976
98	Feldkirch	Vorarlberg	AT63 0100 0000 0557 4988	98-999 / 9982

Remarks:

• The BIC for all tax offices is: BUNDATWW (bank details: BAWAG PSK )
• The tax office number can always be found in the third from last and penultimate digits of the respective tax office IBAN . This relationship can be used for validation purposes. Example: Linz tax office: FA = 46 , IBAN = AT03 0100 0000 0552 4 46 4

"Fictitious" tax account number

In the case of certain legal processes (e.g. complaint fees) it is generally not necessary to state the tax account number assigned to you. In such cases, or if the tax account number has not yet been communicated to you by the tax office, the “fictitious” tax account number provided for this purpose of the respective tax office must be used ( see table in the previous section ).

Check digit

Only a single decimal check digit is used as the checksum for the tax account number .

use cases

There are three situations in which the check digit needs to be calculated:

1. When generating new tax account numbers. In this case, and are given for a specific tax office, and the digits are determined (e.g. using a random number generator ). Then you have to calculate in order to get a complete tax account number . (When assigning it, make sure that the newly generated tax account number does not already exist; each tax account number assigned must be unique.) The tax office normally generates it. ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle N_{1},\dotsc ,N_{6}}$ ${\displaystyle P}$ ${\displaystyle (F,A,N_{1},\dotsc ,N_{6},P)}$
2. When validating - i.e. when checking for formal validity - existing tax account numbers. In this case, the full tax account number is given, and the check digit is recalculated and compared with the given check digit . Validation is recommended wherever tax account numbers are dealt with that could possibly contain errors (e.g. typing , transferring , scanning errors, etc.). ${\displaystyle (F,A,N_{1},\dotsc ,N_{6},P)}$ ${\displaystyle P'}$ ${\displaystyle P}$
3. When reconstructing existing tax account numbers, for which the check digit has not become unequivocally or not at all legible, while all the other eight digits are unequivocally legible.

purpose

The purpose of the check digit is to intentionally add redundancy to the tax account number in order to be able to identify incorrect tax account numbers as incorrect with the highest possible probability . The check digit can be used to ensure that input errors lead to formal errors, which are then reliably recognizable. (The emphasis here is on "can", that is, it can succeed, but does not always have to succeed.)

Ultimately, the aim is to be able to eliminate incorrect tax account numbers and then only work with correct tax account numbers in order to make administration more efficient.

Calculation of the check digit

The check digit of the tax account number can be  calculated as follows - based on the Luhn algorithm : ${\displaystyle P}$ ${\displaystyle (F,A,N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},P)}$

1. ${\displaystyle S:=F+Q(A)+N_{1}+Q(N_{2})+N_{3}+Q(N_{4})+N_{5}+Q(N_{6})}$ With ${\displaystyle Q(z):=q(2\cdot z)={\begin{cases}2\cdot z,&{\text{wenn }}0\leq z\leq 4{\text{,}}\\2\cdot z-9,&{\text{wenn }}5\leq z\leq 9{\text{;}}\end{cases}}}$
2. ${\displaystyle P:=(-S){\bmod {1}}0}$ = (80 - S) % 10

The formula for calculating the check digit is therefore an 8- digit , surjective function with the eight variables : ${\displaystyle P}$ ${\displaystyle F,A,N_{1},\dotsc ,N_{6}}$

${\displaystyle P(F,A,N_{1},\dotsc ,N_{6})=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})]{\bmod {1}}0}$

The following applies here:

• ${\displaystyle q(n)}$ ... cross sum of the number ${\displaystyle n\in \mathbb {N} _{0}}$
• ${\displaystyle 0\leq Q(z)=q(2\cdot z)\leq 9}$ with number ${\displaystyle z\in \{0,\dotsc ,9\}}$
• ${\displaystyle S_{\mathrm {min} }\leq S\leq S_{\mathrm {max} }}$,  ... auxiliary amount ${\displaystyle S\in \mathbb {N} _{0}}$
  • ${\displaystyle S_{\mathrm {min} }=8\cdot 0=0}$ For ${\displaystyle F=A=N_{1}=\dotsb =N_{6}=0}$
  • ${\displaystyle S_{\mathrm {max} }=8\cdot 9=72}$ For ${\displaystyle F=A=N_{1}=\dotsb =N_{6}=9}$
• ${\displaystyle 0\leq S\leq 72}$
• ${\displaystyle 0\leq P\leq 9}$

This is the "mathematical" (and not the "symmetrical") variant of the modulo function . For programming, it is advisable here to always call the modulo function with non-negative numbers, since you then no longer have to worry about whether the programming language used implements the "mathematical" or the "symmetrical" variant, because then both variants deliver the same (and here fitting) results. In the above calculation of the check digit, this is achieved by adding to , which results in the program code . ${\displaystyle a{\bmod {m}}}$ ${\displaystyle 8\cdot m=8\cdot 10=80}$ ${\displaystyle -S}$ (80 - S) % 10

Function graph for for the digit in the interval [0..9] ${\displaystyle Q(z)}$ ${\displaystyle z}$

Function graph for for the value in the interval [0..9] ${\displaystyle z(Q)}$ ${\displaystyle Q}$

Details on the cross-sum calculation of the number multiplied by 2:

Table of values for ${\displaystyle Q(z)}$

Digit ${\displaystyle z}$	${\displaystyle \color {gray}2\cdot z}$	Checksum ${\displaystyle Q(z)=q(2\cdot z)}$
0	00	0
1	02	2
2	04th	4th
3	06th	6th
4th	08th	8th
5	10	1
6th	12	3
7th	14th	5
8th	16	7th
9	18th	9

A corresponding calculation example can be found below.

From this it can be seen that the function is or is bijective . This is why there is also its inverse function , which can be used to unambiguously determine the associated digit for a given value . The inverse function is as follows: ${\displaystyle Q(z)=q(2\cdot z)}$ ${\displaystyle Q\colon \{0,\dotsc ,9\}\to \{0,\dotsc ,9\},z\mapsto q(2\cdot z)}$ ${\displaystyle Q^{-1}}$ ${\displaystyle Q}$ ${\displaystyle z}$

${\displaystyle z(Q)=Q^{-1}(Q)={\begin{cases}{\frac {Q}{2}},&{\text{wenn }}Q{\text{ gerade,}}\\{\frac {Q+9}{2}},&{\text{wenn }}Q{\text{ ungerade;}}\end{cases}}}$ With ${\displaystyle Q\in \{0,\dotsc ,9\}}$

This means that the number can be represented as a function of , i.e. the cross sum of the number multiplied by 2. ${\displaystyle z}$ ${\displaystyle Q}$

Example: If is, then must be; see table or inverse function: ${\displaystyle Q=q(2\cdot z)=5}$ ${\displaystyle z=7}$ ${\displaystyle \textstyle z(5)=Q^{-1}(5)={\frac {5+9}{2}}={\frac {14}{2}}=7}$

This clear reversibility is important so that, for example, the reconstruction of an unknown digit of a tax account number works in all cases.

Validation

Validation based on the structure

A given tax account number is usefully first of all, then to consider whether their building meets the basic requirements of the structure.

Validation based on the check digit

A given tax account number can be checked for formal validity ( data validation , plausibility check ) by comparing its given check digit with the newly calculated check digit for this tax account number . The calculation method is for and the same. ${\displaystyle (F,A,N_{1},\dotsc ,N_{6},P)}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle P}$ ${\displaystyle P'}$

• Agree and agree, the given tax account number is formally valid. (Note: Just because a given tax account number is purely formally valid, it does not necessarily have to be valid before the tax office; it may not have been assigned by the tax office, for example.) ${\displaystyle P}$ ${\displaystyle P'}$
• Agree and do not agree, the given tax account number is certainly invalid. ${\displaystyle P}$ ${\displaystyle P'}$

Two related calculation examples can be found below.

Validation based on the tax office number

To check whether a given tax account number can be valid, the tax office number FA contained in it can be compared with a list of valid tax office numbers (see e.g. the table above or the web links below ).

• If the tax office number FA contained in the tax account number appears in the list of valid tax office numbers, the given tax account number can (but does not have to) be valid before the tax office.
• If, however, the tax office number FA is not included in the list of valid tax office numbers, the given tax account number is not valid in any case.

Error detection

General

Purely formal errors in a tax account number can always be detected with a 100% probability using the check digit. That is, when a charge account number the predetermined formal criteria ( structure not match / structure, in particular check digit), this is seen to be 100%.

The difference is the detection of random input errors , e.g. B. Typing, transmission, scanning errors u. The like. Here, in particular, pressing the wrong key when typing on a keyboard , swapping (adjacent) digits (e.g. 69 instead of 96 ; number rotator ), mixing up digits due to poor legibility (e.g. 8 instead of of 3 ) or poor intelligibility while calling (z. B. 3 instead of 2 ) or the like is meant.

The probability that existing input errors are actually recognizable is 90%. Statistically, random input errors have the effect that in 90% of all possible cases ( ) a tax account number is created which - due to its check digit - is no longer formally valid and can then be reliably (100%) recognized as invalid. ${\displaystyle \textstyle 1-{\frac {10^{8}}{10^{9}}}}$

The probability that the error detection fails, i.e. that existing input errors are wrongly not detected, is 10% here. Statistically, random input errors have the effect that in 10% of all possible cases ( ) a tax account number is created that is formally valid despite the input error and despite its check digit and therefore cannot be recognized as invalid. ${\displaystyle \textstyle {\frac {10^{8}}{10^{9}}}}$

These two probabilities are due to the fact that only a single decimal check digit is used as the checksum, which can only assume one specific value for a specific tax account number out of ten possible digit values.

If a formal error (based on the check digit) was recognized, then it is 100% clear that the tax account number is formally invalid, i.e. it contains errors. However, it cannot be determined where (with which or which of the nine digits) the error is. The exact number of errors cannot be determined either; it is then only clear that at least one of the nine digits must be wrong. An automatic correction of the input error is not possible. So there is no fault tolerance here , just simple fault detection.

More information on recognizing special input errors can be found in the section of the same name below.

Detection of special input errors

In order to get a better impression of which input errors, under which circumstances and with which probabilities, can be recognized as input errors or not recognized as such, the following special cases should be considered more closely, which can be assumed to occur frequently in everyday practice .

Exactly one single incorrect digit

If exactly one single digit is incorrect for a given tax account number (and all other eight digits are correct), then this can always be recognized with a probability of 100%. If you make a mistake on just one of the nine digits, then this can always be recognized without exception.

If you pick any number out of the total of eight digits from the tax account number and assume all the remaining seven digits of the tax account number (as well as their specific check digit ) as fixed (of the original eight variables only one variable remains, and the remaining original variables become constants), then the (in the general case yes) 8-digit, surjective function, according to which the check digit is calculated, becomes ${\displaystyle (F,A,N_{1},\dotsc ,N_{6},P)}$ ${\displaystyle z}$ ${\displaystyle F,A,N_{1},\dotsc ,N_{6}}$ ${\displaystyle P_{z}}$

${\displaystyle P(F,A,N_{1},\dotsc ,N_{6})=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})]{\bmod {1}}0}$,

Always a new, now-digit function , which for is bijective always. Due to the bijectivity of, it follows that every incorrect value of the digit is reflected in a check digit value in the course of the validation that definitely deviates from the given check digit . A typo alone can therefore always be recognized. ${\displaystyle {\tilde {P}}(z)}$ ${\displaystyle z\in \{0,\dotsc ,9\}}$ ${\displaystyle {\tilde {P}}(z)}$ ${\displaystyle z}$ ${\displaystyle {\tilde {P}}(z)}$ ${\displaystyle P_{z}}$

For comparison, the general case: If there are wrong digits in a tax account number , then there is generally no longer any bijectivity of the then -digit function . The probability that existing input errors will actually be recognized is then only 90%. If you wanted to increase this probability, you would have to introduce a multi-digit checksum or expand the base (= 10 in the decimal system ) of the check digit, e.g. B. switch to a hexadecimal check digit. ${\displaystyle k}$ ${\displaystyle 2\leq k\leq 8}$ ${\displaystyle k}$ ${\displaystyle {\tilde {P}}(z_{1},\dotsc ,z_{k})}$

Exactly an interchange of two immediately adjacent digits

If there is an interchange of two immediately adjacent digits and , then this can not be recognized if the following applies: ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$

1. ${\displaystyle z_{1}\neq z_{2}}$ and
2. ${\displaystyle \{[Q(z_{1})+z_{2}]{\bmod {1}}0\}=\{[Q(z_{2})+z_{1}]{\bmod {1}}0\}}$

The first of the two conditions results from the consideration that two digits can only be swapped (in a potentially result-relevant manner) if they are different from one another; a swap of the two digits at z. B. the number 44 does not change anything. The second condition arises from the consideration that the error detection on the basis of the check digit fails if the check digit value of the non-swapped variant does not differ from the swapped one.

These two conditions are met if and only applies. ${\displaystyle z_{1}=0\land z_{2}=9}$ ${\displaystyle z_{1}=9\land z_{2}=0}$

That means:

• Whenever the digit sequence 09 or 90 is contained in a tax account number, a number rotated within this number sequence (here called "09-number rotator" for short) cannot be recognized by validation. So if 09 is mistakenly written instead of 90 correctly , or if 90 is mistakenly written instead of 09 correctly , then these “09-number rotators” cannot be recognized by the check digit.
• All other number rotations between two immediately adjacent digits can always be recognized by validation.

If there is only one twisted number in a given tax account number, it can be recognized with a probability of 98%.

If you wanted to be able to recognize a single number rotated in a given tax account number with 100% probability, you would have to do without all those tax account numbers that contain the sequence 09 or 90 ; d. This means that such tax account numbers should never be assigned by the tax office. Specifically, the tax office would have to do without 12,372,894 of the 10 ⁸ = 100,000,000 possible tax account numbers, i.e. almost 12.4% of the value stock . For example, the “fictitious” tax account numbers of the tax offices ( see table ) show that the tax office generally does not use this additional protection option.

Maybe it can help to avoid "09-Zahlendreher" tends to be when you type of tax account numbers, if present, preferably the (usually remote) numeric keypad of the keyboard uses, instead of using the usual numeric keys. In the case of the numeric keypad, the keys for 0and are 9far apart, while in the case of the usual numeric keys on the keyboard they are directly next to each other and can therefore be more prone to accidental mix-ups. The fact that the 0key on the numeric keypad is usually much wider than all other number keys can - due to the resulting tactile feedback - provide additional, so to speak "intuitive" security when typing.

Reconstruction of an unknown digit

If there is a tax account number that has a single digit, regardless of where, that has not become unequivocally or not at all legible, this unknown digit can be clearly reconstructed. The prerequisites for this are, on the one hand, that all other eight digits are legible without any doubt and, on the other hand, that the original tax account number was formally valid. ${\displaystyle (F,A,N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},P)}$ ${\displaystyle x}$ ${\displaystyle x}$

Attention: Reconstruction is always at your own risk! With such a reconstruction, the property that is normally present (in the case of a completely intact tax account number because of the check digit) that input errors can be recognized is lost. If (random) input errors are made in the course of the reconstruction, then these will not be recognized and the reconstructed digit will generally (more precisely: with a probability of 90%) be wrong. Such a reconstruction should therefore only be carried out with caution and special attention, if at all. From a purely mathematical point of view, however, the reconstructions described here are flawless. In any case, you should first try to obtain the full tax account number (or the unknown number) in other ways. At the end of a reconstruction, it is advisable to subject the completed tax account number to a validation in order to at least identify or exclude any formal errors.

Method 1: direct calculation

The calculation of the unknown number depends on where the number is in the tax account number . The calculation can be done as follows: ${\displaystyle x}$ ${\displaystyle x}$

direct calculation of an unknown digit ${\displaystyle x}$

structure	unknown digit in place of ${\displaystyle x}$	calculation
x A-NNN / NNNP	${\displaystyle F=x}$	${\displaystyle x=F=[-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$
F x -NNN / NNNP	${\displaystyle A=x}$	${\displaystyle Q_{A}=Q(A)=[-F-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$ ${\displaystyle x=A=z(Q_{A})}$
FA- x NN / NNNP	${\displaystyle N_{1}=x}$	${\displaystyle x=N_{1}=[-F-Q(A)-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$
FA-N x N / NNNP	${\displaystyle N_{2}=x}$	${\displaystyle Q_{N_{2}}=Q(N_{2})=[-F-Q(A)-N_{1}-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$ ${\displaystyle x=N_{2}=z(Q_{N_{2}})}$
FA-NN x / NNNP	${\displaystyle N_{3}=x}$	${\displaystyle x=N_{3}=[-F-Q(A)-N_{1}-Q(N_{2})-Q(N_{4})-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$
FA-NNN / x NNP	${\displaystyle N_{4}=x}$	${\displaystyle Q_{N_{4}}=Q(N_{4})=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-N_{5}-Q(N_{6})-P]{\bmod {1}}0}$ ${\displaystyle x=N_{4}=z(Q_{N_{4}})}$
FA-NNN / N x NP	${\displaystyle N_{5}=x}$	${\displaystyle x=N_{5}=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-Q(N_{6})-P]{\bmod {1}}0}$
FA-NNN / NN x P	${\displaystyle N_{6}=x}$	${\displaystyle Q_{N_{6}}=Q(N_{6})=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-P]{\bmod {1}}0}$ ${\displaystyle x=N_{6}=z(Q_{N_{6}})}$
FA-NNN / NNN x	${\displaystyle P=x}$	${\displaystyle x=P=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-Q(N_{6})]{\bmod {1}}0}$

1. ↑ For the sake of completeness and better comparison possibilities, the check digit formula is again given here in compact form. It may well happen that the check digit needs to be reconstructed.

The following applies here:

• ${\displaystyle Q(z):=q(2\cdot z)}$ with number ${\displaystyle z\in \{0,\dotsc ,9\}}$
• ${\displaystyle q(n)}$ ... cross sum of the number ${\displaystyle n\in \mathbb {N} _{0}}$
• ${\displaystyle z(Q)=Q^{-1}(Q)={\begin{cases}{\frac {Q}{2}},&{\text{wenn }}Q{\text{ gerade,}}\\{\frac {Q+9}{2}},&{\text{wenn }}Q{\text{ ungerade;}}\end{cases}}}$ With ${\displaystyle Q\in \{0,\dotsc ,9\}}$

Two related calculation examples can be found below.

Method 2: iterative calculation

The unknown number can also be calculated using an iterative method . You can proceed as follows: ${\displaystyle x}$

• If the unknown digit takes the place of the check digit , it can be reconstructed directly, i.e. non-iteratively, using the check digit formula. ${\displaystyle x}$ ${\displaystyle P}$
• For all other cases: You assign different values ​​from the set to the unknown digit until the newly calculated check digit matches the given check digit . Usually you will assign the values ​​0, 1, 2,…, 9 one after the other, but you can also choose any other order. In the best case, on the first attempt, generally after an average of five, in any case after a maximum of ten attempts, one has then found the desired value for by trying out the ten different possibilities. ${\displaystyle x}$ ${\displaystyle \{0,\dotsc ,9\}}$ ${\displaystyle P'}$ ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle x}$

A corresponding calculation example can be found below.

Note: If digits are unknown for a given tax account number , but the check digit itself is known, then the tax account number can no longer be clearly reconstructed purely from the check digit. Under certain circumstances, clarity can still be achieved by using other criteria; however, in general this will not work. For example, potential tax account number candidates can be excluded if their (calculated) tax office number FA does not appear in the list of tax office numbers actually used ( see table ). In any case, by appropriately generalizing the iterative procedure outlined above, i.e. by trying out all possible variations , all those potential candidates can be identified that would result in a formally valid tax account number. In practice, however, this is not recommended. ${\displaystyle k}$ ${\displaystyle P}$ ${\displaystyle 2\leq k\leq 8}$ ${\displaystyle k=2}$ ${\displaystyle 10^{k}}$

Examples

Example 1 - Generating a new tax account number

The tax account number for the tax office no. 98 (Feldkirch) should be generated, whereby the first six digits of the tax number should be 123456 .

So the following is given:

• ${\displaystyle F=9,A=8}$
• ${\displaystyle N_{1}=1,N_{2}=2,N_{3}=3,N_{4}=4,N_{5}=5,N_{6}=6}$

The check digit is sought : ${\displaystyle P}$

• ${\displaystyle {\begin{aligned}S&=F+q(2\cdot A)+N_{1}+q(2\cdot N_{2})+N_{3}+q(2\cdot N_{4})+N_{5}+q(2\cdot N_{6})\\&=9+\underbrace {q(\overbrace {2\cdot 8} ^{16})} _{7}+1+\underbrace {q(\overbrace {2\cdot 2} ^{4})} _{4}+3+\underbrace {q(\overbrace {2\cdot 4} ^{8})} _{8}+5+\underbrace {q(\overbrace {2\cdot 6} ^{12})} _{3}\\&=9+7+1+4+3+8+5+3=40\end{aligned}}}$
• ${\displaystyle P=(-S){\bmod {1}}0=(-40){\bmod {1}}0=0}$= (80 - S) % 10= (80 - 40)% 10 = 40% 10 = 0

The tax account number is therefore: 98-123 / 4560

Example 2 - Validation of a given tax account number

The tax account number 90-123 / 4567 should be checked for formal validity.

So the following is given:

• ${\displaystyle F=9,A=0}$
• ${\displaystyle N_{1}=1,N_{2}=2,N_{3}=3,N_{4}=4,N_{5}=5,N_{6}=6}$
• ${\displaystyle P=7}$

The check digit is to be calculated , which is then to be compared with : ${\displaystyle P'}$ ${\displaystyle P}$

• ${\displaystyle {\begin{aligned}S&=F+q(2\cdot A)+N_{1}+q(2\cdot N_{2})+N_{3}+q(2\cdot N_{4})+N_{5}+q(2\cdot N_{6})\\&=9+\underbrace {q(\overbrace {2\cdot 0} ^{0})} _{0}+1+\underbrace {q(\overbrace {2\cdot 2} ^{4})} _{4}+3+\underbrace {q(\overbrace {2\cdot 4} ^{8})} _{8}+5+\underbrace {q(\overbrace {2\cdot 6} ^{12})} _{3}\\&=9+0+1+4+3+8+5+3=33\end{aligned}}}$
• ${\displaystyle P'=(-S){\bmod {1}}0=(-33){\bmod {1}}0=7}$= (80 - S) % 10= (80 - 33)% 10 = 47% 10 = 7

The given tax account number 90-123 / 4567 is formally valid because it is here . ${\displaystyle P=P'}$

Example 3 - Validation of a given tax account number

The tax number 987/6543 , which should be assigned to the tax office “Vienna 02/20/21/22” ( FA = 12 ), should be checked for formal validity. (The tax account number 12-987 / 6543 must therefore be checked for formal validity.)

So the following is given:

• ${\displaystyle F=1,A=2}$
• ${\displaystyle N_{1}=9,N_{2}=8,N_{3}=7,N_{4}=6,N_{5}=5,N_{6}=4}$
• ${\displaystyle P=3}$

The check digit is to be calculated , which is then to be compared with : ${\displaystyle P'}$ ${\displaystyle P}$

• ${\displaystyle {\begin{aligned}S&=F+q(2\cdot A)+N_{1}+q(2\cdot N_{2})+N_{3}+q(2\cdot N_{4})+N_{5}+q(2\cdot N_{6})\\&=1+\underbrace {q(\overbrace {2\cdot 2} ^{4})} _{4}+9+\underbrace {q(\overbrace {2\cdot 8} ^{16})} _{7}+7+\underbrace {q(\overbrace {2\cdot 6} ^{12})} _{3}+5+\underbrace {q(\overbrace {2\cdot 4} ^{8})} _{8}\\&=1+4+9+7+7+3+5+8=44\end{aligned}}}$
• ${\displaystyle P'=(-S){\bmod {1}}0=(-44){\bmod {1}}0=6}$= (80 - S) % 10= (80 - 44)% 10 = 36% 10 = 6

The given tax account number 12-987 / 6543 is formally not valid because it is here . ${\displaystyle P\neq P'}$

Example 4 - direct reconstruction of an unknown digit of a tax account number

The tax account number 46-376 / 5 x 21 is to be reconstructed, the penultimate digit x has become unrecognizable.

So the following is given:

• ${\displaystyle F=4,A=6}$
• ${\displaystyle N_{1}=3,N_{2}=7,N_{3}=6,N_{4}=5,N_{6}=2}$
• ${\displaystyle P=1}$

Is wanted . ${\displaystyle N_{5}=x}$

The calculation can be done as follows:

${\displaystyle {\begin{aligned}x&=N_{5}=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-Q(N_{6})-P]{\bmod {1}}0\\&=[-4-\underbrace {q(\overbrace {2\cdot 6} ^{12})} _{3}-3-\underbrace {q(\overbrace {2\cdot 7} ^{14})} _{5}-6-\underbrace {q(\overbrace {2\cdot 5} ^{10})} _{1}-\underbrace {q(\overbrace {2\cdot 2} ^{4})} _{4}-1]{\bmod {1}}0\\&=[-4-3-3-5-6-1-4-1]{\bmod {1}}0=[-27]{\bmod {1}}0=3\end{aligned}}}$

The full tax account number is therefore 46-376 / 5 3 21 . A final validation confirms the correctness of the result.

Example 5 - direct reconstruction of an unknown digit of a tax account number

The tax account number 03-826 / 15 x 4 is to be reconstructed, the penultimate digit x has become unrecognizable.

So the following is given:

• ${\displaystyle F=0,A=3}$
• ${\displaystyle N_{1}=8,N_{2}=2,N_{3}=6,N_{4}=1,N_{5}=5}$
• ${\displaystyle P=4}$

Is wanted . ${\displaystyle N_{6}=x}$

The calculation can be done as follows:

• ${\displaystyle {\begin{aligned}Q_{N_{6}}&=Q(N_{6})=[-F-Q(A)-N_{1}-Q(N_{2})-N_{3}-Q(N_{4})-N_{5}-P]{\bmod {1}}0\\&=[-0-\underbrace {q(\overbrace {2\cdot 3} ^{6})} _{6}-8-\underbrace {q(\overbrace {2\cdot 2} ^{4})} _{4}-6-\underbrace {q(\overbrace {2\cdot 1} ^{2})} _{2}-5-4]{\bmod {1}}0\\&=[0-6-8-4-6-2-5-4]{\bmod {1}}0=[-35]{\bmod {1}}0=5\end{aligned}}}$
• ${\displaystyle x=N_{6}=z(Q_{N_{6}})=z(5)=\underbrace {\tfrac {5+9}{2}} _{{\text{da }}Q_{N_{6}}=5{\text{ ungerade}}}=7}$

The full tax account number is therefore 03-826 / 15 7 4 . A final validation confirms the correctness of the result.

Example 6 - iterative reconstruction of an unknown digit of a tax account number

The tax account number 54-2 x 7/9451 is to be reconstructed, the fourth digit (counting from the left) of which x has become unrecognizable.

So the following is given:

• ${\displaystyle F=5,A=4}$
• ${\displaystyle N_{1}=2,N_{3}=7,N_{4}=9,N_{5}=4,N_{6}=5}$
• ${\displaystyle P=1}$

Is wanted . ${\displaystyle N_{2}=x}$

The iterative calculation can be carried out as follows:

Run number	selected value for ${\displaystyle x}$	temporary tax account number	calculated check digit ${\displaystyle P'}$	Validation result (is the same ?) ${\displaystyle P'}$ ${\displaystyle P}$
1	0	54-2 0 7/945 1	4th	No
2	1	54-2 1 7/945 1	2	No
3	2	54-2 2 7/945 1	0	No
4th	3	54-2 3 7/945 1	8th	No
5	4th	54-2 4 7/945 1	6th	No
6th	5	54-2 5 7/945 1	3	No
7th	6th	54-2 6 7/945 1	1	Yes

In this example, in the seventh iteration step, the digit was found which results in a successful validation of the temporary tax account number. The number found completes the given tax account number. The full tax account number is therefore 54-2 6 7/9451 . A final validation confirms the correctness of the result.

See also

• Identifier
• tax number
• Tax office

Web links

• TIN on-the-Web - Online test module for tax identification numbers (Taxpayer Identification Numbers, TIN) of the European Commission

Individual evidence

1. ↑ ^{a b c} BMF - tax payments (tax office payment). In: bmf.gv.at. Retrieved December 12, 2016 . 
2. ↑ ^{a b c} BMF - Other organizational measures. In: bmf.gv.at. Retrieved December 12, 2016 . 
3. ^ BMF - Tax & Customs Forms. In: service.bmf.gv.at. Retrieved December 12, 2016 . 
4. ↑ ^{a b c} Tax identification numbers (Austria). (PDF; 196 kB) European Commission, January 19, 2017, accessed December 9, 2017 . 
5. ↑ ^{a b} Tax office payment in MBS / XML. (PDF; 217 kB) STUZZA GesmbH, December 12, 2016, accessed on December 9, 2017 . 
6. ^ BMF - Offices & Authorities. In: service.bmf.gv.at. Retrieved December 12, 2016 . 
7. ↑ List of tax offices. (CSV; approx. 30 kB) Federal Ministry of Finance, accessed on December 22, 2016 (the list is regenerated by the tax office every time it is accessed to ensure that it is up to date). 
8. ↑ BMF - tax payments (information on the tax account number). In: bmf.gv.at. Retrieved December 12, 2016 . 
9. ↑ simple program for counting the "09-number rotators"; it returns 12372894 as the result
    Source code of a computer program in the Java programming language for counting the "09-number rotators"

   ---

   import java.lang.Math;
   
   public class Count_09_Pairs {
   
     public static void main(String[] args) {
       long from = 0;
       long to   = 99999999;
   
       if ((from > to) || (from < 0) || (to > 99999999)) {
         System.out.println("falsche Intervall-Angaben");
         System.exit(0);
       }
   
       int lg_to = (int)(Math.log10((double)(to + 1)));
   
       long number;
       long z0, z1, z2, z3, z4, z5, z6, z7, z8;
       long count = 0;
   
       for (number = from; number <= to; ++number) {
         z0 = number % 10;
         z1 = ((long)(number / 10)) % 10;
         z2 = ((long)(number / 100)) % 10;
         z3 = ((long)(number / 1000)) % 10;
         z4 = ((long)(number / 10000)) % 10;
         z5 = ((long)(number / 100000)) % 10;
         z6 = ((long)(number / 1000000)) % 10;
         z7 = ((long)(number / 10000000)) % 10;
         z8 = ((long)(number / 100000000)) % 10;
   
         if ( ( (lg_to >= 2) && ( ((z0 == 0) && (z1 == 9)) || ((z0 == 9) && (z1 == 0)) ) ) ||
              ( (lg_to >= 3) && ( ((z1 == 0) && (z2 == 9)) || ((z1 == 9) && (z2 == 0)) ) ) ||
              ( (lg_to >= 4) && ( ((z2 == 0) && (z3 == 9)) || ((z2 == 9) && (z3 == 0)) ) ) ||
              ( (lg_to >= 5) && ( ((z3 == 0) && (z4 == 9)) || ((z3 == 9) && (z4 == 0)) ) ) ||
              ( (lg_to >= 6) && ( ((z4 == 0) && (z5 == 9)) || ((z4 == 9) && (z5 == 0)) ) ) ||
              ( (lg_to >= 7) && ( ((z5 == 0) && (z6 == 9)) || ((z5 == 9) && (z6 == 0)) ) ) ||
              ( (lg_to >= 8) && ( ((z6 == 0) && (z7 == 9)) || ((z6 == 9) && (z7 == 0)) ) ) ||
              ( (lg_to >= 9) && ( ((z7 == 0) && (z8 == 9)) || ((z7 == 9) && (z8 == 0)) ) ) ) {
           count++;
           /* System.out.println("" +
                              ((lg_to >= 9) ? z8 : "") +
                              ((lg_to >= 8) ? z7 : "") +
                              ((lg_to >= 7) ? z6 : "") +
                              ((lg_to >= 6) ? z5 : "") +
                              ((lg_to >= 5) ? z4 : "") +
                              ((lg_to >= 4) ? z3 : "") +
                              ((lg_to >= 3) ? z2 : "") +
                              ((lg_to >= 2) ? z1 : "") +
                              ((lg_to >= 1) ? z0 : "") +
                              " (" + number + ")" +
                              " => 09-/90-Paar gefunden: " + count); */
         }
       }
   
       System.out.println("Gesamtanzahl an 09-/90-Paaren in [" + from + ".." + to + "]: " + count);
     }
   
   }
   

Please note the information on legal issues!