added Teichmueller representatives in Zp

data/P_adic_numbers/Teichmueller_representatives_in_Zp/generate.sage

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+import yaml
+import os
+import mpmath
+
+path = 'data/P_adic_numbers/Teichmueller_representatives_in_Zp/'
+
+prec10 = 30 #relative precision in base 10
+
+p_range = prime_range(20)
+
+RIFprec = RealIntervalField(prec10 * 3.4 * 2)
+
+numbers = {}
+for p in p_range:
+	print("p:",p)
+	numbers_p = {}
+
+	prec_p = ceil(30*log(10,p))
+	Q_p = Qp(p, prec=prec_p, print_mode='val-unit')
+
+	if p == 2:
+		k_range = [1,-1]
+		Ts = Q_p.roots_of_unity()
+		assert(Ts[0] == 1)
+	else:
+		k_range = [1..p-1]
+		Ts = Q_p.teichmuller_system()
+
+	for i, k in enumerate(k_range):
+		number = Ts[i]
+		assert((number - k).valuation() > 0)
+
+		number_str = str(number)
+		numbers_p[str(k)] = number_str
+
+	numbers[str(p)] = numbers_p
+
+filename = os.path.join(path, 'numbers.yaml')
+yaml.dump(numbers, stream = open(filename, 'w'), sort_keys = False)

data/P_adic_numbers/Teichmueller_representatives_in_Zp/numbers.yaml

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+'2':
+  '1': 1 + O(2^100)
+  '-1': 1267650600228229401496703205375 + O(2^100)
+'3':
+  '1': 1 + O(3^63)
+  '2': 1144561273430837494885949696426 + O(3^63)
+'5':
+  '1': 1 + O(5^43)
+  '2': 620681016124000438335084264557 + O(5^43)
+  '3': 516187361092159859058714563568 + O(5^43)
+  '4': 1136868377216160297393798828124 + O(5^43)
+'7':
+  '1': 1 + O(7^36)
+  '2': 919754687174599885029200936914 + O(7^36)
+  '3': 919754687174599885029200936915 + O(7^36)
+  '4': 1731976158685053586749822444686 + O(7^36)
+  '5': 1731976158685053586749822444687 + O(7^36)
+  '6': 2651730845859653471779023381600 + O(7^36)
+'11':
+  '1': 1 + O(11^29)
+  '2': 822308133844989680537587352955 + O(11^29)
+  '3': 572261535344934988362615426173 + O(11^29)
+  '4': 1226189280167312456547755556038 + O(11^29)
+  '5': 610166615504233810041653075634 + O(11^29)
+  '6': 976142681667257764372783629257 + O(11^29)
+  '7': 360120017004179117866681148853 + O(11^29)
+  '8': 1014047761826556586051821278718 + O(11^29)
+  '9': 764001163326501893876849351936 + O(11^29)
+  '10': 1586309297171491574414436704890 + O(11^29)
+'13':
+  '1': 1 + O(13^27)
+  '2': 1116618631499798074920340780077 + O(13^27)
+  '3': 975827136128499513407176961479 + O(13^27)
+  '4': 975827136128499513407176961480 + O(13^27)
+  '5': 605937130343626045281406468488 + O(13^27)
+  '6': 662510823181559912916642767669 + O(13^27)
+  '7': 530022469330932103642552240448 + O(13^27)
+  '8': 586596162168865971277788539629 + O(13^27)
+  '9': 216706156383992503152018046637 + O(13^27)
+  '10': 216706156383992503152018046638 + O(13^27)
+  '11': 75914661012693941638854228040 + O(13^27)
+  '12': 1192533292512492016559195008116 + O(13^27)
+'17':
+  '1': 1 + O(17^25)
+  '2': 1748142213128282398726840189819 + O(17^25)
+  '3': 4507271044819188044594632079357 + O(17^25)
+  '4': 1734790125262563614608664379008 + O(17^25)
+  '5': 1108547014373663604814057093494 + O(17^25)
+  '6': 3690023658820840516759985642774 + O(17^25)
+  '7': 5172556674846629984057735564060 + O(17^25)
+  '8': 894230316526990233257745210857 + O(17^25)
+  '9': 4876397095821412145681824780200 + O(17^25)
+  '10': 598070737501772394881834426997 + O(17^25)
+  '11': 2080603753527561862179584348283 + O(17^25)
+  '12': 4662080397974738774125512897563 + O(17^25)
+  '13': 4035837287085838764330905612049 + O(17^25)
+  '14': 1263356367529214334344937911700 + O(17^25)
+  '15': 4022485199220119980212729801238 + O(17^25)
+  '16': 5770627412348402378939569991056 + O(17^25)
+'19':
+  '1': 1 + O(19^24)
+  '2': 4009161444819018765352579221678 + O(19^24)
+  '3': 4450382154506130333807703844861 + O(19^24)
+  '4': 480289528207326051041876330690 + O(19^24)
+  '5': 3560780668364302281443987788618 + O(19^24)
+  '6': 3931280912843706139881466679323 + O(19^24)
+  '7': 3227183485951824626020576406997 + O(19^24)
+  '8': 3227183485951824626020576406998 + O(19^24)
+  '9': 487192489909814626792952267908 + O(19^24)
+  '10': 4411570441051032190923343010013 + O(19^24)
+  '11': 1671579445009022191695718870923 + O(19^24)
+  '12': 1671579445009022191695718870924 + O(19^24)
+  '13': 967482018117140677834828598598 + O(19^24)
+  '14': 1337982262596544536272307489303 + O(19^24)
+  '15': 4418473402753520766674418947231 + O(19^24)
+  '16': 448380776454716483908591433060 + O(19^24)
+  '17': 889601486141828052363716056243 + O(19^24)
+  '18': 4898762930960846817716295277920 + O(19^24)

data/P_adic_numbers/Teichmueller_representatives_in_Zp/table.yaml

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+ID: INPUT{id.yaml}
+
+Title: >
+  Teichmüller representatives in $\mathbb{Z}_p$
+
+Definition: >
+  Let $p$ be a rational prime,
+  let $q=p$ for $p>2$ and $q=4$ for $p=2$,
+  let $G = (\mathbb{Z}/q\mathbb{Z})^\times$, 
+  and let $\omega: G \to \mathbb{Z}_p^*$ be
+  the Teichmüller character CITE{Wiki}.
+  The images $\omega(k)$ of elements $k \in G$ are their
+  Teichmüller representatives in $\mathbb{Z}_p$.  
+
+Parameters:
+  p:
+    type: Z
+    constraints: prime
+
+  k:
+    type: Z
+    constraints: 
+    - > 
+      $1 \leq k < p$ for $p>2$
+    - >
+      $k = \pm 1$ for $p=2$
+    
+Comments:
+  comment-roots-of-unity: >
+    The set of Teichmüller representatives in $\mathbb{Z}_p$
+    equals the set of non-zero roots of unity in $\mathbb{Z}_p$.
+    The $k$'th Teichmüller representative reduces to $k$ modulo $p$.
+  
+Formulas: 
+
+Programs: 
+
+References:
+
+Links:
+  Wiki:
+    title: "Wikipedia: Teichmüller character"
+    url: https://en.wikipedia.org/wiki/Teichm%C3%BCller_character
+
+Similar tables:
+
+Keywords:
+
+Tags:
+- p-adic
+- zeros
+
+Data properties:
+  type: Qp
+  complete: no
+
+Display properties:
+  number-header: Teichmüller representative of $k$ in $\mathbb{Z}_p$
+
+Numbers: INPUT{numbers.yaml}