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Unless some other LHS value is shown, each identity equates to [1], i.e. PI/4 (see the 'Identity lists' page for a full description of how identities are displayed).

TWO-TERM IDENTITIES

Note: In this context, both cotangent values must be integers

1.85113  *MACHIN (1706)
   4[5]-[239]

3.27920  *MACHIN (1706)     (¢¢6.60113)
   2[3]+[7]
  Eliminated:[2]

4.50522  HERMANN (1706)     (¢¢6.60113)
   2[2]-[7]
Also known to *MACHIN (1706). 
Obtainable by substituting N=2 in the standard 2-term expansion of [N/2]!

5.41783  *MACHIN (1706)
   [2]+[3]

Note: Størmer's proof (in French) that the above four are the ONLY 2-term identities can be inspected at: http://www.numdam.org/item?id=BSMF_1899__27__160_1.

*New attributions to Machin arise from paper by Tweddle (1991) and references therein.

THREE-TERM IDENTITIES

Note: In this context, all three cotangent values must be integers

1.78661  GAUSS (1863)       (¢¢2.64766,¢¢2.89392)
   12[18]+8[57]-5[239]
  Eliminated:{A}

1.78921  SIMSON (1723) (previously attributed to KLINGENSTIERNA (1730))
   8[10]-[239]-4[515]
  Eliminated:{A}([10])

1.93314  STØRMER (1896)
   4[5]-2[478]+[54608393]

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2.03740  WRENCH (1938)
   5[7]+4[53]+2[4443]
  Eliminated:[2]([53])

2.06095  STØRMER (1896)
   4[5]-[240]-[57361]

2.06176  STØRMER (1896)
   4[5]-[238]+[56883]

2.07473  STØRMER (1896)
   4[5]-[241]-[28800]
  Eliminated:([5])[15&16]

2.07639  STØRMER (1896)
   4[5]-[237]+[28322]
  Eliminated:[2&9&37]([5])

2.09728  STØRMER (1896)     (¢¢2.89392)
   6[8]+2[57]+[239]
  Eliminated:{A}

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2.11091  STØRMER (1896)
   4[5]-2[577]-[1393]
  Eliminated:[2&14]([5])

2.11989  STØRMER (1896)
   4[5]-[252]-[4633]

2.13179  STØRMER (1896)
   4[5]-2[408]+[1393]
  Eliminated:[2&14]([5])

2.13182  STØRMER (1896)
   4[5]-[226]+[4155]

2.13862  STØRMER (1896)
   4[5]-[265]-[2436]
  Eliminated:[2&6&27]([5])

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2.16394  STØRMER (1896)
   4[5]-[213]+[1958]
  Eliminated:[2&23&136]([5])

2.17326  STØRMER (1896)
   6[7]-4[57]+[239]
  Eliminated:{A}

2.17589  STØRMER (1896)
   4[5]-[408]-[577]
  Eliminated:[2&14]([5])

2.19184  EULER  (1779?)
   5[7]+2[26]-2[2057] (published 1798 in the form  5[7] + 2[79/3])
  Eliminated:[2]([26])

2.21212  STØRMER (1896)
   5[7]+2[27]+2[1068]
  Eliminated:[2]([27])

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2.21251  STØRMER (1896)
   5[7]+4[68]+2[117]
  Eliminated:[2][6]

2.25218  STØRMER (1896)
   5[7]+2[28]+2[443]
  Eliminated:[2]([28])

2.27900  STØRMER (1896)
   5[7]+4[43]-2[117]
  Eliminated:[2][6]

2.31431  STØRMER (1896)
   5[6]-[68]-3[117]
  Eliminated:[2]([6])

2.32440  STØRMER (1896)
   8[8]-4[18]+3[239]
  Eliminated:{A}

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2.34119  STØRMER (1896)
   5[7]+2[43]+2[68]
  Eliminated:[2]][6]

2.36012  STØRMER (1896)
   5[7]+2[23]-2[182]
  Eliminated:[2]([23]])

2.37608  STØRMER (1896)
   4[6]+4[31]-[239]
  Eliminated:[5]([6])

2.38081  STØRMER (1896)
   5[6]-[43]-2[117]
  Eliminated:[2]([6])

2.40039  STØRMER (1896)
   4[7]+4[18]-[239]
  Eliminated:{A}

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2.44299  STØRMER (1896)
   5[6]-3[43]+2[68]
  Eliminated:[2]([6])

2.47347  SCHELLBACH (1832)
   5[8]+2[18]+3[57]
  Eliminated:{A}

2.47375  EULER (1738 or earlier - published 1744)
   4[5]-[70]+[99]
  Eliminated:[2&12]([5])

2.49643  STØRMER (1896)
   3[7]+4[11]-2[682]
  Eliminated:[2]([11])
Note: 2[11] - [682] = [11/2], but (1/Log11 + 1/Log682) < 1/Log(11/2).
From N=15 onwards 1/Log(N/2) < {1/LogN +1/Log(N(N²+3)/2)}.

2.54945  STØRMER (1896)
   5[7]+2[18]-2[57]
  Eliminated:{A}

2.55182  STØRMER (1896)
   4[5]-[41]+2[99]
  Eliminated:[2&12]([5])

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2.59270  STØRMER (1896)
   4[5]+[41]-2[70]
  Eliminated:[2&12]([5])
Note: 2[5] - [70] = [5/2], but (1/Log5 + 1/Log70) < 1/Log(5/2).
From N=15 onwards 1/Log(N/2) < {1/LogN +1/Log(N(N²+3)/2)}.

2.71106  STØRMER (1896)
   2[7]+4[8]+[239]
  Eliminated:{A}

2.71400  STØRMER (1896)
   3[7]+4[13]+2[38]
  Eliminated:[2][4]

2.73282  *MACHIN (1706)
   3[4]+[20]+[1985]
*See note under 2-term identities, above.

2.79683  STØRMER (1896)
   5[5]-3[18]-2[57]
  Eliminated:[2]([5])

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2.83772  STØRMER (1896)
   4[4]-4[21]-[239]
  Eliminated:{A}([4])

2.95000  WRENCH (1938)
2[1]=5[3]-2[53]-[4443]

2.95191  STØRMER (1896)
   4[6]+[7]-2[117]
  Eliminated:[2]([6])

3.01409  STØRMER (1896)
   2[6]+3[7]+2[68]
  Eliminated:[2]([6])

3.08587  STØRMER (1896)
5[1]=12[3]+4[57]-[239]
  Eliminated:{A}

3.08724  SCHELLBACH (1832)
   3[7]+2[8]+2[18]
  Eliminated:{A}

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3.10445  STØRMER (1896)
2[1]=5[3]-[26]+[2057]

3.10750  STØRMER (1896)
   2[5]+3[8]+[57]
  Eliminated:[2]([5])

3.12473  STØRMER (1896)
2[1]=5[3]-[27]-[1068]

3.12512  STØRMER (1896)
2[1]=5[3]-2[68]-[117]
  Eliminated:[2][6]

3.16478  STØRMER (1896)
2[1]=5[3]-[28]-[443]
  Eliminated:[2]([28])

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3.18349  STØRMER (1896)
   2[5]+3[7]-2[57]
  Eliminated:[2]([5])

3.19161  STØRMER (1896)
2[1]=5[3]-2[43]+[117]
  Eliminated:[2][6]

3.19167  STØRMER (1896)
   3[4]+[13]-[38]
  Eliminated:[2]([4])

3.25380  STØRMER (1896)
2[1]=5[3]-[43]-[68]
  Eliminated:[2][6]

3.27273  STØRMER (1896)
2[1]=5[3]-[23]+[182]
  Eliminated:[2]([23])

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3.28648  STØRMER (1896)
   2[3]+2[14]-[1393]
  Eliminated:[2]([14])

3.31299  STØRMER (1896)
   3[1]=8[3]-4[18]+[239]
  Eliminated:{A}

3.33463  STØRMER (1896)
   3[5]+2[8]-[18]
  Eliminated:[2]([5])

3.40904  STØRMER (1896)
   3[3]-2[11]+[682]
  Eliminated:[2]([11])

3.46206  STØRMER (1896)
2[1]=5[3]-[18]+[57]
  Eliminated:{A}

3.47726  STØRMER (1896)
   4[4]-[7]-2[38]
  Eliminated:[2]([4])

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3.52866  STØRMER (1896)
   2[3]+2[17]+[41]
  Eliminated:

3.62366  STØRMER (1896)
   4[3]-4[8]-[239]
  Eliminated:{A}

3.62661  STØRMER (1896)
   3[3]-2[13]-[38]
  Eliminated:[2][4]

3.64258  STØRMER (1896)
   2[3]+2[12]-[41]
  Eliminated:[2]([12])

3.72128  STØRMER (1896)
   2[5]+[7]+2[8]
  Eliminated:[2]([5])

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3.74197  STØRMER (1896)
   2[4]+[7]+2[13]
  Eliminated:[2]([4])

3.77273  STØRMER (1896)
   2[3]+[8]+[57]
  Eliminated:{A}

3.80824  STØRMER (1896)
   2[3]+[9]+[32]
  Eliminated:[2]([9])

3.83524  STØRMER (1896)
   2[3]+[12]+[17]
  Eliminated:[2]([12])

3.92670  STØRMER (1896)
   3[3]-[6]-[68]
  Eliminated:[2]([6])

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3.99319  STØRMER (1896)
   2[3]+[6]-[43]
  Eliminated:[2]([6])

3.99985  STØRMER (1896)
   3[3]-[8]-[18]
  Eliminated:{A}

4.09610  STØRMER (1896)
   3[3]-[5]+[57]
  Eliminated:[2]([5])

4.17603  WRENCH (1938)
3[1]=5[2]+2[53]+[4443]

4.31190  STØRMER (1896)
7[1]=12[2]-4[57]+[239]
  Eliminated:([2])[5]

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4.32322  STØRMER (1896)
   2[3]+[5]-[18]
  Eliminated:[2]([5])

4.33048  STØRMER (1896)
3[1]=5[2]+[26]-[2057]

4.35075  STØRMER (1896)
3[1]=5[2]+[27]+[1068]

4.35114  STØRMER (1896)
3[1]=5[2]+2[68]+[117]
  Eliminated:([2])[6]

4.38987  STØRMER (1896)
   [3]+2[4]-[38]
  Eliminated:[2]([4])

4.39081  STØRMER (1896)
3[1]=5[2]+[28]+[443]

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4.41764  STØRMER (1896)
3[1]=5[2]+2[43]-[117]
  Eliminated:

4.47982  STØRMER (1896)
3[1]=5[2]+[43]+[68]
  Eliminated:([2])[6]

4.49875  STØRMER (1896)
3[1]=5[2]+[23]-[182]

4.51250  STØRMER (1896)
   2[2]-2[14]+[1393]

4.53902  STØRMER (1896)
5[1]=8[2]+4[18]-[239]
  Eliminated:([2])[5]

4.63507  STØRMER (1896)
2[1]=3[2]+2[11]-[682]

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4.68808  STØRMER (1896)
3[1]=5[2]+[18]-[57]
  Eliminated:([2])[5]

4.75469  STØRMER (1896)
   2[2]-2[17]-[41]
  Eliminated:([2])[12]

4.84969  STØRMER (1896)
3[1]=4[2]+4[8]+[239]
  Eliminated:([2])[5]

4.85264  STØRMER (1896)
2[1]=3[2]+2[13]+[38]
  Eliminated:([2])[4]

4.86860  STØRMER (1896)
   2[2]-2[12]+[41]

4.99875  STØRMER (1896)
   2[2]-[8]-[57]
  Eliminated:([2])[5]

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5.03427  STØRMER (1896)
   2[2]-[9]-[32]

5.06127  STØRMER (1896)
   2[2]-[12]-[17]

5.09054  STØRMER (1896)
   [2]+2[6]-[117]

5.15273  STØRMER (1896)
2[1]=3[2]+[6]+[68]

5.21922  STØRMER (1896)
   2[2]-[6]+[43]

5.22588  STØRMER (1896)
2[1]=3[2]+[8]+[18]
  Eliminated:([2])[5]

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5.32212  STØRMER (1896)
2[1]=3[2]+[5]-[57]

5.54924  STØRMER (1896)
   2[2]-[5]+[18]

5.85991  STRASSNITZKY (1844)
   [2]+[5]+[8]

5.88060  STØRMER (1896)
   [2]+[4]+[13]

6.60113  HCL (1993)
   3[2]-[3]-2[7]
  Note: any multiple of [2]-[3]-[7] may be added to the 
  last expression without affecting its LHS value ([1])

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