Module:Number
此模块的文档可以在Module:Number/doc创建
local getArgs = require('Module:Arguments').getArgs
local p = { PrimeTable = {} }
local numdata = nil
--[[
輸入一個數字n，檢查n是否為半完全數，如是列出最先搜到的和為自身之因數數列，否則輸出空陣列 (一般Lua函數)
注意 : 此為指數時間複雜問題
--最差情況時間複雜度為 O(2^因數個數)
--呼叫時請謹慎
]]
function p._checkSemiperfectNumber(input)
local number = tonumber(input) or 0
if p.PrimeTable.table_max == nil then p.PrimeTable = require('Module:Factorization') end
local divs_all = p.PrimeTable._findDivisor(number)
return p._checkSemiperfectNumberByDivisor(divs_all)
end
--[[
輸入因數列表檢查數字是否為半完全數，如是列出最先搜到的和為自身之因數數列，否則輸出空陣列 (一般Lua函數)
注意 : 此為指數時間複雜問題
--最差情況時間複雜度為 O(2^因數個數)
--呼叫時請謹慎
]]
function p._checkSemiperfectNumberByDivisor(input)
if xpcall(function()_=#input pairs(input) end,function(_)end) == true then
local flattern = function(a_table)
local flattern_result = {}
if xpcall(function()_=#a_table pairs(a_table) end,function(_)end) == true then
for key,value in pairs(a_table) do
if value == 1 then flattern_result[#flattern_result + 1] = key end
end
table.sort(flattern_result)
end
return flattern_result
end
local number = input[#input]
local divs_all = input
local divs = {}
local result = {}
local divs_subset = require("Module:Combination").getCombinationGenerator()
local sum = 0
for i = 1, #divs_all do if divs_all[i] ~= number then
divs[divs_all[i]] = 1
sum = sum + divs_all[i]
result[divs_all[i]] = 1
end end
if sum < number then return {} end
if sum == number then return flattern(result) end
divs_subset:init(divs, 0)
local calc_semi = sum - number
local add_buffer = 0
for i = 1, (#divs_all)-1 do
divs_subset.count = i
divs_subset:set({nil})
divs_subset:fill()
local middle_flag = true;
while middle_flag == true do
local inside_flag = true;
while inside_flag == true do
add_buffer=0
for j=1,#(divs_subset.list) do
add_buffer = add_buffer + divs_subset.list[j]
end
if calc_semi - add_buffer == 0 then
for j=1,#(divs_subset.list) do
result[divs_subset.list[j]] = nil
end
return flattern(result)
end
inside_flag = divs_subset:next()
end
local backtrack_flag = true;
while backtrack_flag == true do
local delete_num = divs_subset.list[#(divs_subset.list)]
divs_subset.number_count[delete_num] = divs_subset.number_count[delete_num] - 1
divs_subset.list[#(divs_subset.list)] = nil
if #(divs_subset.list) <= 0 then break end
middle_flag = divs_subset:next()
divs_subset:fill()
backtrack_flag = (#(divs_subset.list) < divs_subset.count) or #(divs_subset.list) <= 0
end
if #(divs_subset.list) <= 0 then break end
end
end
end
return {}
end
--[[
檢查一個數字是否為半完全數，如是列出最先搜到的和為自身之因數數列，否則不輸出 (支援#invoke:)
注意 : 此為指數時間複雜問題
--最差情況時間複雜度為 O(2^因數個數)
--呼叫時請謹慎
]]
function p.checkSemiperfectNumber(frame)
local args
if frame == mw.getCurrentFrame() then
-- We're being called via #invoke. The args are passed through to the module
-- from the template page, so use the args that were passed into the template.
args = getArgs(frame) --frame.args
else
-- We're being called from another module or from the debug console, so assume
-- the args are passed in directly.
args = frame
end
return tostring(table.concat(p._checkSemiperfectNumber(args[1] or args['1']),','))
end
--[[
列出一個或多個整數與因數相關的性質 (支援#invoke:)
運算速度較快，適合進行大量輸出，如1000個以下的數字
]]
function p.numberDivisorInformation(frame)
-- For calling from #invoke.
local args
local can_math = false
local yesno = require('Module:Yesno')
if frame == mw.getCurrentFrame() then
-- We're being called via #invoke. The args are passed through to the module
-- from the template page, so use the args that were passed into the template.
args = getArgs(frame, {parentFirst=true}) --frame.args
can_math = yesno(args['use math'] or args['use_math'])
SemiperfectNumber = yesno(args['SemiperfectNumber'] or args['semiperfectNumber'] or
args['Semiperfect Number'] or args['semiperfect Number'] or args['Semiperfect_Number'] or args['semiperfect_Number'] or 'yes')
else
-- We're being called from another module or from the debug console, so assume
-- the args are passed in directly.
args = frame
end
local body = ''
local print_information = function(result_str, number, page, name, display, linkQ, bullet, describe, head_describe)
local head_text = args[tostring(number) .. (name or '數字')] or args[tostring(number) .. (page or '數字')] or
args[tostring(number) .. (display or '數字')]
local tail_text = args[tostring(number) .. (page or '數字') .. "end"] or args[tostring(number) .. (page or '數字') .. "f"] or
args[tostring(number) .. (name or '數字') .. "end"] or args[tostring(number) .. (name or '數字') .. "f"] or
args[tostring(number) .. (display or '數字') .. "end"] or args[tostring(number) .. (display or '數字') .. "f"]
result_str = result_str .. bullet .. (head_text or '')
result_str = result_str .. (head_describe or '')
if linkQ == true then
result_str = result_str .. "[[" .. page
if (display ~= nil) and (display ~= '') then result_str = result_str .. '|' .. display end
result_str = result_str .. "]]"
else
if (display ~= nil) and (display ~= '') then
result_str = result_str .. display
else
result_str = result_str .. name
end
end
if (describe ~= nil) and (describe ~= '') then
result_str = result_str .. '，' .. describe
end
if tail_text ~= nil then result_str = result_str .. '，' .. tail_text end
result_str = result_str .. '。\n'
return result_str
end
--預設為只處理參數1的數字
local num_start = args[1] or args['1']
local num_end = args[1] or args['1']
--讀取範圍
if args['start'] then num_start = args['start'] end
if args['end'] then num_end = args['end'] end
local sort_invert = yesno(args['sort invert'] or args['sort_invert'] or false)
--範圍內只有1個數字
local one_number = false
if num_start == num_end then one_number = true end
--建立白名單
local white_list = {}
local white_index = {}
local white_list_enable = false
if args['white_list'] or args['white list'] then
white_list = mw.text.split((args['white_list'] or args['white list']),',')
white_list_enable = true
end
--建立白名單查表
for first,second in pairs(white_list) do
if tonumber(second) then
white_index[tonumber(second)] = 1
end
end
--未輸入視為預設，從零開始
num_start = tonumber(num_start or 0)
num_end = tonumber(num_end or 0)
if mw.ustring.lower(tostring(num_start)) == 'nan' or mw.ustring.lower(tostring(num_end)) == 'nan' then
return require("Module:Error").error({[1]="錯誤：\'" .. (args[1] or args['1'] or args['start'] or args['end']) .. '\' '
.. '不是一個數-{}-字。'
})
end
--超出可處理範圍
if p.PrimeTable.table_max == nil then p.PrimeTable = require('Module:Factorization') end
if (num_start or 0) > p.PrimeTable.limit or (num_end or 0) > p.PrimeTable.limit
or (num_start == nil) or (num_end == nil)
then
return require("Module:Error").error({[1]="錯誤：無法處理數-{}-字\'" .. (args[1] or args['1'] or args['start'] or args['end']) .. '\''})
end
if (num_end or 0) < (num_start or 0) then
return require("Module:Error").error({[1]="錯誤：範圍不正確"})
end
--輸入太多數字
if (num_end or 0) - (num_start or 0) > 1510 then
if(#white_list > 1510) then
--無論是計算時間或者模板展開大小都有超出的風險
return require("Module:Error").error({[1]="錯誤：數字太多"})
end
end
local high_time_complexity = false
local print_number_count = num_end - num_start + 1
if white_list_enable then print_number_count = math.min( print_number_count, #white_list ) end
--開始列印數字與因數分解相關的性質
local number = tonumber(args[1] or args['1'] or 0)
local number_step=1
if sort_invert==true then
local swap_temp = num_start
number_step = -1
num_start = num_end
num_end = swap_temp
end
for number = num_start , num_end , number_step do
--數字位於白名單內
if(not white_list_enable or white_index[number] == 1) then
if one_number == false then
--如果數字很多建立章節 (用[[#數字]]可跳到的章節)
body = body .. ';' .. tostring(number) .. '<span id=\"' .. tostring(number) .. '\">\n'
end
if args[tostring(number)..'_head'] or args[tostring(number)..' head'] then
--列印位於章節頭部的資訊，如主條目
local str = string.gsub((args[tostring(number)..'_head'] or args[tostring(number)..' head']) , "\n", "\n:")
body = body .. str .. '\n'
end
if math.abs(math.floor(number) - number) > 0 then
body = body .. p._simpleRealText(number, can_math, frame) .. '\n'
else
if p.PrimeTable.table_max == nil then p.PrimeTable = require('Module:Factorization') end
--因數分解
local primedata = p.PrimeTable._factorization(number)
local divdata = p.PrimeTable._findDivisorByPrimeFactor(primedata)
--因數分解發生錯誤，停止執行並返回錯誤
if primedata.has_err ~= nil or divdata.has_err ~= nil then
local Error = require("Module:Error")
if primedata.has_err ~= nil then return body .. '\n' .. Error.error({[1]="錯誤：" .. primedata.has_err})
else return body .. '\n' .. Error.error({[1]="錯誤：" .. divdata.has_err}) end
end
local is_prime = true --表示數字是否為質數
local num_sqrt = math.sqrt(number) --表示數字的平方根
local admirable_div = nil --表示佩服數的相減因數
local is_unusual = nil --表示數字是否有大於平方根的質因數
local div_sum = 0 --因數和
local prime_count = 0 --相異質因數個數
local div_sum_harmonic = 0 --因數調和總和
local div_harmonic_avg = 0.001 --因數調和平均數
local div_count = 0 --因數數量
local exp_num = 0 --質因數總指數 (Ω)
local prime_digits = 0
local buffer_text = ''
--掃描質因數
for first,second in pairs(primedata) do
if first ~= 'has_err' then
if first ~= 1 and first ~= number then
is_prime = false
--判斷是否為不尋常數 (是否有大於平方根的質因數)
if first > num_sqrt then is_unusual = first end
end
if first ~= 1 then
--計算質因數總指數 (Ω)
exp_num = exp_num + second
--計算相異質因數數量
prime_count = prime_count + 1
prime_digits = prime_digits +
mw.ustring.len( mw.ustring.format( "%d",math.floor(math.abs(first)) ) )
if second > 1 then prime_digits = prime_digits +
mw.ustring.len( mw.ustring.format( "%d",math.floor(math.abs(second)) ) ) end
end
end
end
--計算因數和
for first,second in pairs(divdata) do
if first ~= 'has_err' then
if second ~= number then div_sum = div_sum + second end
div_sum_harmonic = div_sum_harmonic + (1.0 / second)
div_count = div_count + 1
end
end
--計算因數的調和平均數
div_harmonic_avg = (div_count + 0.000000) / div_sum_harmonic
for first,second in pairs(divdata) do
if first ~= 'has_err' then
--判斷數字是否為佩服數
if div_sum - second * 2 == number then admirable_div = second end
end
end
--產生<math>格式，以印出質因數分解於條目
local times = " x "
local pow_h = "<sup>"
local pow_f = "</sup>"
if can_math then
times = "\\times "
pow_h = "^{"
pow_f = "} "
end
--印出質因數分解於條目
local factors_str = p.PrimeTable.create_factorization_string(primedata, times, pow_h, pow_f)
if can_math then factors_str = frame:callParserFunction{name = "#tag:math", args = {factors_str}} end
--開始列印整數性質
if is_prime == true then
--質因數分解只有一個元素的情況
if number == 0 then
body = body .. '*[[0|中性數]]。\n'
elseif number == 1 then
body = body .. '*[[单位數字|單位數]]。\n'
elseif number == -1 then
body = body .. '*[[負數]][[单位數字|單位]]。\n'
else
--是個質數
body = body .. '*'
if p.PrimeTable.lists[number] ~= nil then
body = body .. '第' .. tostring(p.PrimeTable.lists[number]) .. '個'
end
body = body .. '[[質數]]。\n'
if p.PrimeTable.table_max == nil then p.PrimeTable = require('Module:Factorization') end
--查質數表，找前一個和下一個質數
local next_p = p.PrimeTable._nextPrime(number)
local last_p = p.PrimeTable._lastPrime(number)
if number > 7919 then high_time_complexity = true end
if tonumber(next_p) ~= nil or tonumber(last_p) ~= nil then
--判斷是否為孿生質數
if (next_p or number) - number == 2 or number - (last_p or number) == 2 then
body = body .. '**' .. (args[tostring(number) .. "孿生質數"] or '')
body = body .. "[[孪生素数]]，"
local is_printted = false
if tonumber(next_p) ~= nil then
if (next_p or number) - number == 2 then
is_printted = true
body = body .. '為（' .. number ..'、 [[' .. next_p ..']]）'
end
if number - (last_p or number) == 2 then
if is_printted == true then body = body .. '以及'
else body = body .. '為' end
body = body .. '（[[' .. last_p ..']]、 ' .. number ..'）'
end
body = body .. '\n'
end
end
end
end
else
--列印所有因數和質因數
local div_string = ''
body = print_information(body, number, number < 0 and "负数" or "合数", number < 0 and "負數" or "合數", nil, true, '*', (args["因數說明h"] or ("[[因數|"..(number < 0 and "" or "正").."因數]]有")) ..
table.concat( divdata, '、', 1, #divdata - 1 ) ..'和' .. tostring(divdata[#divdata]) .. (args["因數說明f"] or '') )
body = print_information(body, number, "整数分解", "整數分解", "質因數分解", true, "*:", factors_str)
if number > 2 then
--列印純粹因數和相關性質 (虧數/完全數/過剩數)
if div_sum ~= number then
local tail_str=''
if div_sum < number then
body = body .. '*' .. (args[tostring(number) .. "虧數"] or '')
body = body .. '[[亏数]]'
tail_str = "，虧度為" .. tostring(number-div_sum)
if args[tostring(number) .. "虧數f"] ~= nil then tail_str = tail_str .. '，' .. args[tostring(number) .. "虧數f"] end
else
body = body .. '*' .. (args[tostring(number) .. "過剩數"] or '')
body = body .. '[[过剩数]]'
tail_str = "，盈度為" .. tostring(div_sum-number)
if args[tostring(number) .. "過剩數f"] ~= nil then tail_str = tail_str .. '，' .. args[tostring(number) .. "過剩數f"] end
end
body = body .. '，[[:w:真因數和|真因數和]]為' .. tostring(div_sum) .. tail_str .. '\n'
else
body = print_information(body, number, "完全数", "完全數", nil, true, '*')
end
if SemiperfectNumber and print_number_count <= 1010 and div_sum >= number then --過剩數才需探討是否半完全或奇異
--最差情況時間複雜度為 O(2^因數個數)
--因此根據要做的數字數量決定是否放棄計算
if (#divdata <= 12 or (print_number_count <= 505 and #divdata <= 14)
or (print_number_count <= 200 and #divdata <= 16) or (print_number_count <= 10 and #divdata <= 20)
or (print_number_count <= 2 and #divdata <= 25) or (print_number_count <= 1 and #divdata <= 32) ) then
--此為指數時間複雜度演算法，因此當數字量大於505時則放棄計算
local simi_div = p._checkSemiperfectNumberByDivisor(divdata)
if #simi_div > 0 then
buffer_text = ''
if div_sum ~= number then
buffer_text = "和為本身的其中一組因數為[[" .. tostring( table.concat(simi_div, "]]、 [[") ) .. "]]"
end
body = print_information(body, number, "半完全数", "半完全數", nil, true, "**", buffer_text )
else
body = print_information(body, number, "奇異數_(數論)", "奇异数", "奇異數", true, "**")
end
if #divdata > 16 then high_time_complexity = true end
end
end
--列印與因數平均數相關性質
if math.abs(div_harmonic_avg - math.floor(div_harmonic_avg)) <= 1e-6 then
body = print_information(body, number, "歐爾調和數", "欧尔调和数", nil, true, '*',
"因數[[调和平均数]]為" .. string.format("%d",math.floor(div_harmonic_avg)))
end
--列印與質因數相關性質
if is_unusual ~= nil then
body = print_information(body, number, "不尋常數", "不寻常数", nil, true, '*',
"大於平方根的質因數為" .. tostring(is_unusual) )
end
if exp_num == 2 then
body = print_information(body, number, "半素数", "半質數", nil, true, '*' )
end
if admirable_div ~= nil then
body = print_information(body, number, "佩服數", "佩服数", nil, true, '*',
"佩服[[因數]]為" .. tostring(admirable_div) )
end
--列印無平方數因數的數之相關性質
if exp_num == prime_count then
body = print_information(body, number,
"无平方数因数的数", "無平方數因數的數", nil, true, '*' )
if prime_count == 3 then
body = print_information(body, number, "楔形数", "楔形數", nil, true, "**" )
end
end
end
--列印自然數相關性質
if number > -1 then
if math.abs(num_sqrt - math.floor(num_sqrt)) <= 1e-6 then
body = print_information(body, number, "平方数", "平方數", nil, true, '*',
"為" .. string.format("%d", math.floor(num_sqrt)) .. "的平方" )
end
end
end
if number > -1 and number ~=1 then
if math.abs(math.ceil(num_sqrt) - math.floor(num_sqrt)) > 1e-6 and math.abs(math.ceil(num_sqrt) * math.floor(num_sqrt) - number) <= 1e-6 then
body = print_information(body, number, "普洛尼克数", "普洛尼克數", nil, true, '*',
"為" .. mw.ustring.format( "%d", math.floor(num_sqrt) ) .."與"
.. mw.ustring.format( "%d", math.ceil(num_sqrt) ) .."的乘積" )
end
if is_prime ~= true then
local num_digts = mw.ustring.len( mw.ustring.format( "%d",math.floor(math.abs(number)) ) )
if prime_digits == num_digts then
body = print_information(body, number, "等數位數", "等数位数", nil, true, '*', nil, "[[十进制]]的")
elseif prime_digits < num_digts then
body = print_information(body, number, "節儉數", "节俭数", nil, true, '*', nil, "[[十进制]]的")
elseif prime_digits > num_digts then
body = print_information(body, number, "奢侈數", "奢侈数", nil, true, '*', nil, "[[十进制]]的")
end
end
end
end
--列印其他性質
if args[number] or args[tostring(number)] then
local str = (args[number] or args[tostring(number)])
body = body .. str .. '\n'
end
end
end
if high_time_complexity == true then body = body .. '[[Category:使用高時間複雜度演算法的條目]]' end
return mw.text.trim(body)
end
--用於少量數字(十幾個) 性質輸出 (支援客製化)
function p.manyNumberInformation(frame)
numdata = require("Module:Number/data")
local can_math = false
local args
if frame == mw.getCurrentFrame() then
-- We're being called via #invoke. The args are passed through to the module
-- from the template page, so use the args that were passed into the template.
local yesno = require('Module:Yesno')
args = getArgs(frame, {readOnly = false, parentFirst=true}) --frame.args
can_math = yesno(args['use math'] or args['use_math'])
else
-- We're being called from another module or from the debug console, so assume
-- the args are passed in directly.
args = frame
end
if _flag_mass_output == nil then _flag_mass_output = true end
local count = tonumber(args['count']) or 1
local number = args[1] or args['1']
local body = ''
if number == nil then return '' end
for i = number,number+count+1 do
if count > 1 then
--如果數字很多建立章節 (用[[#數字]]可跳到的章節)
body = body .. ';' .. tostring(i) .. '<span id=\"' .. tostring(i) .. '\">\n'
end
if args[tostring(i)..'_head'] or args[tostring(i)..' head'] then
--列印位於章節頭部的資訊，如主條目
local str = string.gsub((args[tostring(i)..'_head'] or args[tostring(i)..' head']) , "\n", "\n:")
body = body .. str .. '\n'
end
if(can_math)then if args[1] then args[1] = i frame.args[1] = i end if args['1'] then args['1'] = i frame.args['1'] = i end
body = body .. p.singleNumberInformation(frame) .. '\n'
else if args[1] then args[1] = i end if args['1'] then args['1'] = i end
body = body .. p.singleNumberInformation(args) .. '\n'
end
--列印其他性質
if args[i] or args[tostring(i)] then
local str = (args[i] or args[tostring(i)])
body = body .. str .. '\n'
end
end
body = mw.text.trim(body)
if _flag_use_user_define_string == true then body = body .. "[[Category:使用自定義Module:Number語句的頁面]]" end
return body
end
function p._checkAddSubtract(num_str)
local body = ''
local real, imag = 0, 0
local split_str = mw.text.split(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(
num_str or '',
'%s+',''),'%++([%d%.])',',+%1'),'%++([ij])',',+1%1'),'%-+([%d%.])',',-%1'),'%-+([ij])',',-1%1'),'%*+([%d%.])',',*%1'),'%*+([ij])',',*1%1'),'%/+([%d%.])',',/%1'),'%/+([ij])',',/1%1'),',')
local first = true
local continue = false
for k,v in pairs(split_str) do
continue = false
local val = mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.ustring.gsub(mw.text.trim(v),'[ij]+','i'),'^(%.)','0%1'),'^([%d%.])','+%1'),'([%+%-])([%d%.])','%1\48%2'),'^([ij])','+1%1')
if mw.ustring.find(val,"%/") or mw.ustring.find(val,"%*") then return end
if val == nil or val == '' then if first == true then first = false continue = true else return end end
if not continue then
local num_text = mw.ustring.match(val,"[%+%-][%d%.]+i?")
if num_text ~= val then return end
local num_part = tonumber(mw.ustring.match(num_text,"[%+%-][%d%.]+"))
if num_part == nil then return end
if mw.ustring.find(num_text,"i") then
imag = imag + num_part
else
real = real + num_part
end
end
end
return real, imag
end
function p._checkInf(num_str)
local inf_siring = {'inf','infinitas','infinitas','infinity','無限','无限','无限大','無限大','無窮','无穷','无穷大','無窮大','∞','-∞','+∞',
'正無限','正无限','正无限大','正無限大','正無窮','正无穷','正无穷大','正無窮大',
'負無限','負无限','負无限大','負無限大','負無窮','負无穷','負无穷大','負無窮大'
}
for k,v in pairs(inf_siring) do
if num_str == v then
return tonumber("inf")
end
end
end
function p._simpleRealText(number, can_math, frame)
local body = ''
body = '*' .. tostring(number) .. '是一個[[实数]]。\n'
if number < 0 then
body = body .. '*' .. '[[负数]]。\n'
else
body = body .. '*' .. '[[正數]]。\n'
end
if math.abs(math.floor(1 / number) - 1 / number) <= 1e-15 then -- 要求超高精度
body = body .. '*' .. tostring(1 / number) .. '的[[倒数]]。\n'
end
if math.abs(math.floor(number * number) - number * number) <= 1e-15 then -- 要求超高精度
body = body .. '*' .. tostring(number * number) .. '的[[平方根]]。\n'
end
if math.abs(number - math.pi) <= 1e-15 then -- 要求超高精度
body = body .. '*[[圓周率]]。\n'
end
return body
end
function p._simpleComplexText(real, imag, can_math, frame)
local body = ''
if cmath==nil then cmath = require("Module:Complex Number").cmath.init() end
local the_number = cmath.getComplexNumber(real, imag)
local length = cmath.abs(the_number)
local argument = cmath.arg(the_number)
local complex_str, gfactors_str = tostring(the_number), ""
local sq_len = math.sqrt(length)
local sq_real, sq_imag = sq_len * math.cos(argument/2.0), sq_len * math.sin(argument/2.0)
local sq_num, sq_str = cmath.getComplexNumber(sq_real, sq_imag), ""
if can_math then complex_str = frame:callParserFunction{name = "#tag:math", args = {complex_str}} end
body = '*' .. complex_str .. '是一個[[复数_(数学)|複數]]。\n'
if numdata._is_integer(real) and numdata._is_integer(imag) then
body = body ..'*' .. '[[高斯整數]]。\n'
local is_gprime = cmath.is_prime_quadrant1(the_number) or cmath.is_prime_quadrant1(-the_number) or
cmath.is_prime_quadrant1(cmath.getComplexNumber(0, 1) * the_number) or
cmath.is_prime_quadrant1(cmath.getComplexNumber(0, -1) * the_number)
if is_gprime ~= nil then
if not is_gprime then
local gprimedata, gfactors = p.PrimeTable._gaussianFactorization(tostring(the_number)), {}
--產生<math>格式，以印出質因數分解於條目
local times = " x "
local pow_h = "<sup>"
local pow_f = "</sup>"
local left_ = "("
local right_ = ")"
if can_math then
times = "\\times "
pow_h = "^{"
pow_f = "} "
left_ = "\\left( "
right_ = "\\right) "
end
for first,second in pairs(gprimedata) do
if first ~= 'has_err' then
local complex_num = cmath.toComplexNumber(first)
if complex_num.real ~= 0 and complex_num.imag ~= 0 then
gfactors[left_ .. tostring(first) .. right_] = second
else
gfactors[first] = second
end
end
end
--印出質因數分解於條目
gfactors_str = p.PrimeTable.create_factorization_string(gfactors, times, pow_h, pow_f)
if mw.text.trim(gfactors_str) == '' then is_gprime = true end
if can_math then
gfactors_str = frame:callParserFunction{name = "#tag:math", args = {gfactors_str}}
end
end
end
if is_gprime then body = body ..'*' .. '[[高斯整數#作为唯一分解整环|高斯質數]]。\n'
elseif length > 1 then body = body ..'*' .. '[[高斯整數#作为唯一分解整环|高斯]][[合数]]。\n'
body = body ..'*:' .. '[[高斯整數#作为唯一分解整环|高斯]][[整数分解]]為，' .. gfactors_str ..'。\n'
end
end
if numdata._is_integer(sq_real) and numdata._is_integer(sq_imag) then
sq_str = tostring(sq_num)
if can_math then sq_str = frame:callParserFunction{name = "#tag:math", args = {sq_str}} end
body = body ..'*' .. '[[平方數]]，為' .. tostring(sq_str) .. '的平方。\n'
end
if numdata._is_integer(length) then
body = body ..'*' .. '[[絕對值]]為' .. tostring(length) .. '。\n'
else
body = body ..'*' .. '[[絕對值]]約為' .. string.format("%.4f",length) .. '。\n'
end
local deg = math.deg(argument)
if numdata._is_integer(deg) then
body = body ..'*' .. '[[辐角]]為' .. tostring(deg) .. '[[度 (角)|度]]。\n'
else
body = body ..'*' .. '[[辐角]]約為' .. string.format("%.4f",deg) .. '[[度 (角)|度]]。\n'
end
return body
end
--用於單一數字性質輸出 (支援客製化)
function p.singleNumberInformation(frame)
if numdata == nil then numdata = require("Module:Number/data") end
-- For calling from #invoke.
local args
local can_math, SemiperfectNumber = false, true
local all_list = {"負數","自然數","整數","質數","孿生質數","梅森質數","高斯質數","合數","質因數分解","高斯整數分解","虧數",
"過剩數","完全數","半完全數","本原半完全數","奇異數","歐爾調和數","高合成數","不尋常數","半質數","佩服數","無平方數因數的數",
"楔形數","平方數","立方數","普洛尼克數","佩爾數","斐波那契數","階乘","質數階乘","自我數","哈沙德數","全哈沙德數","等數位數",
"節儉數","奢侈數","史密夫數","不可及數","可作圖多邊形"};
local print_list, addit_print_list, print_prop_list, print_black_list, black_list = {}, {}, {}, {}, {"自然數","整數","高斯整數分解"};
if frame == mw.getCurrentFrame() then
-- We're being called via #invoke. The args are passed through to the module
-- from the template page, so use the args that were passed into the template.
args = getArgs(frame) --frame.args
local yesno = require('Module:Yesno')
can_math = yesno(args['use math'] or args['use_math'])
SemiperfectNumber = yesno(args['SemiperfectNumber'] or args['semiperfectNumber'] or
args['Semiperfect Number'] or args['semiperfect Number'] or args['Semiperfect_Number'] or args['semiperfect_Number'] or 'yes')
else
-- We're being called from another module or from the debug console, so assume
-- the args are passed in directly.
args = frame
end
if args['other_print_list'] or args['other print list'] then
addit_print_list = mw.text.split(args['other_print_list'] or args['other print list'] or '',',')
end
if args['print_list'] or args['print list'] then
print_list = mw.text.split(args['print_list'] or args['print list'] or '',',')
end
if args['print_black_list'] or args['print black list'] then
print_black_list = mw.text.split(args['print_black_list'] or args['print black list'] or '',',')
end
local print_list_table, print_list_array = {}, {}
for k,v in pairs(print_list) do local val_iter = mw.text.trim(v) if val_iter ~= '' then print_list_table[val_iter] = 1 end end
for k,v in pairs(print_list_table) do print_list_array[#print_list_array + 1] = k end
if #print_list_array <= 0 then
for k,v in pairs(all_list) do local val_iter = mw.text.trim(v) if val_iter ~= '' then print_list_table[val_iter] = 1 end end
end
for k,v in pairs(print_black_list) do
local val_iter = mw.text.trim(v)
if val_iter ~= '' then
print_list_table[val_iter] = nil
end
end
for k,v in pairs(black_list) do
local val_iter = mw.text.trim(v)
if val_iter ~= '' then
print_list_table[val_iter] = nil
end
end
for k,v in pairs(addit_print_list) do
local val_iter = mw.text.trim(v)
if val_iter ~= '' then
print_list_table[val_iter] = 1
end
end
if #print_list_array > 0 then
for k,v in pairs(print_list) do
local val_iter = mw.text.trim(v)
if (tonumber(print_list_table[val_iter]) or 0) == 1 then
print_prop_list[#print_prop_list + 1] = val_iter
print_list_table[val_iter] = print_list_table[val_iter] + 1
end
end
else
for k,v in pairs(all_list) do
local val_iter = mw.text.trim(v)
if (tonumber(print_list_table[val_iter]) or 0) == 1 then
print_prop_list[#print_prop_list + 1] = val_iter
print_list_table[val_iter] = print_list_table[val_iter] + 1
end
end
end
local body, number = '', tonumber(args[1] or args['1'])
--例外處理 : NaN因平方根不存在，造成短除法，除質數無窮迴圈
if mw.ustring.lower(tostring(number)) == 'nan' then
return require("Module:Error").error({[1]="錯誤：\'" .. (args[1] or args['1'] or args['start'] or args['end']) .. '\' '
.. '不是一個數-{}-字。'
})
end
--例外處理 : 預期內的數字，但不應送進短除法做因數分解的數字
local real, imag = (number or 0), 0
if number == nil then
local test_str = mw.ustring.lower(mw.text.trim(args[1] or args['1']))
number = p._checkInf(test_str)
if number == nil then
real,imag = p._checkAddSubtract(test_str)
if real~=nil and imag~=nil then
number = math.sqrt(real*real+imag*imag)
end
end
--例外處理 : 不是數字的東西，直接告知錯誤
if number == nil then
return require("Module:Error").error({[1]="錯誤：「" .. (args[1] or args['1'] or args['start'] or args['end']) .. '」'
.. '不是一個數-{}-字。'
})
end
end
--超出可處理範圍
if p.PrimeTable.table_max == nil then p.PrimeTable = require('Module:Factorization') end
if (number or 0) > p.PrimeTable.limit or -(number or 0) > p.PrimeTable.limit then
return require("Module:Error").error({[1]="錯誤：無法處理數-{}-字「" .. (args[1] or args['1'] or args['start'] or args['end']) .. '」，'
.. '其[[絕對值]]已超出支援的處理範圍（35184372088831）。'
})
end
if imag ~= 0 then return p._simpleComplexText(real, imag, can_math, frame) end
--例外處理 : 非整數的數字被送進短除法，當然在根號n內不會整除，因此會被當成質數，所以不能將非整數的數字被送進短除法做因數分解
if math.abs(math.floor(number) - number) > 0 then
return p._simpleRealText(number, can_math, frame)
end
--------------------------------------------
----- 例外處理結束，開始短除法因數分解 -----
--------------------------------------------
--因數分解
local primedata = p.PrimeTable._factorization(number)
local divdata = p.PrimeTable._findDivisorByPrimeFactor(primedata)
--因數分解發生錯誤，停止執行並返回錯誤
if primedata.has_err ~= nil or divdata.has_err ~= nil then
local Error = require("Module:Error")
if primedata.has_err ~= nil then return body .. '\n' .. Error.error({[1]="錯誤：" .. primedata.has_err})
else return body .. '\n' .. Error.error({[1]="錯誤：" .. divdata.has_err}) end
end
--產生<math>格式，以印出質因數分解於條目
local times = " x "
local pow_h = "<sup>"
local pow_f = "</sup>"
local left_ = "("
local right_ = ")"
if can_math then
times = "\\times "
pow_h = "^{"
pow_f = "} "
left_ = "\\left( "
right_ = "\\right) "
end
local is_gprime = true
local gfactors_str = ''
if tonumber(print_list_table["高斯整數分解"] or -1) >= 1 then
if cmath==nil then cmath = require("Module:Complex Number").cmath.init() end
local cnum = cmath.toComplexNumber(tostring(number))
is_gprime = cmath.is_prime_quadrant1(cnum)
if is_gprime ~= nil then
if not is_gprime then
local gprimedata, gfactors = p.PrimeTable._gaussianFactorization(tostring(number)), {}
for first,second in pairs(gprimedata) do
if first ~= 'has_err' then
local complex_num = cmath.toComplexNumber(first)
if complex_num.real ~= 0 and complex_num.imag ~= 0 then
gfactors[left_ .. tostring(first) .. right_] = second
else
gfactors[first] = second
end
end
end
--印出質因數分解於條目
gfactors_str = p.PrimeTable.create_factorization_string(gfactors, times, pow_h, pow_f)
if mw.text.trim(gfactors_str) == '' then is_gprime = true end
if can_math then
gfactors_str = frame:callParserFunction{name = "#tag:math", args = {gfactors_str}}
end
end
end
end
local is_prime = true --表示數字是否為質數
local num_sqrt = math.sqrt(number) --表示數字的平方根
local admirable_div = nil --表示佩服數的相減因數
local is_unusual = nil --表示數字是否有大於平方根的質因數
local div_sum = 0 --因數和
local prime_count = 0 --相異質因數個數
local div_sum_harmonic = 0 --因數調和總和
local div_harmonic_avg = 0.001 --因數調和平均數
local div_count = 0 --因數數量
local exp_num = 0 --質因數總指數 (Ω)
local prime_digits = 0
local buffer_text = ''
local mersenne_prime_for_perfect_number = ''
--掃描質因數
for first,second in pairs(primedata) do
if first ~= 'has_err' then
if first ~= 1 and first ~= number then
is_prime = false
end
if first ~= 1 then
--判斷是否為不尋常數 (是否有大於平方根的質因數)
if first > num_sqrt then is_unusual = first end
--計算質因數總指數 (Ω)
exp_num = exp_num + second
--計算相異質因數數量
prime_count = prime_count + 1
prime_digits = prime_digits +
mw.ustring.len( mw.ustring.format( "%d",math.floor(math.abs(first)) ) )
if second > 1 then prime_digits = prime_digits +
mw.ustring.len( mw.ustring.format( "%d",math.floor(math.abs(second)) ) ) end
end
end
end
--計算因數和
for first,second in pairs(divdata) do
if first ~= 'has_err' then
if second ~= number then div_sum = div_sum + second end
div_sum_harmonic = div_sum_harmonic + (1.0 / second)
div_count = div_count + 1
end
end
--計算因數的調和平均數
div_harmonic_avg = (div_count + 0.000000) / div_sum_harmonic
for first,second in pairs(divdata) do
if first ~= 'has_err' then
--判斷數字是否為佩服數
if div_sum - second * 2 == number then admirable_div = second end
end
end
--印出質因數分解於條目
local factors_str = p.PrimeTable.create_factorization_string(primedata, times, pow_h, pow_f)
if can_math then
factors_str = frame:callParserFunction{name = "#tag:math", args = {factors_str}}
if div_sum == number then
mersenne_prime_for_perfect_number= "2^{" .. tostring(primedata[2]) .. "}\\times\\left( 2^{" ..
tostring(primedata[2]+1) .. "}-1\\right)"
mersenne_prime_for_perfect_number = frame:callParserFunction{name = "#tag:math", args = {mersenne_prime_for_perfect_number}}
end
end
local simi_div = {}
if div_sum > number and number > 0 then
if SemiperfectNumber and #divdata <= 32 then
--此為指數時間複雜度演算法，因此當因數超過32個時則放棄計算
simi_div = p._checkSemiperfectNumberByDivisor(divdata)
end
end
local printer = numdata.getIntegerInfoPrinter()
printer:init({
number=number,
is_prime=is_prime,
admirable_div=admirable_div,
is_unusual=is_unusual,
div_sum=div_sum,
prime_count=prime_count,
div_sum_harmonic=div_sum_harmonic,
div_harmonic_avg=div_harmonic_avg,
div_count=div_count,
exp_num=exp_num,
prime_digits=prime_digits,
divdata=divdata,
primedata=primedata,
simi_div=simi_div,
factors_str=factors_str,
mersenne_prime_for_perfect_number=mersenne_prime_for_perfect_number,
is_gprime=is_gprime,
gfactors_str=gfactors_str,
});
local text = ''
for k,v in pairs(print_prop_list) do
if _flag_use_user_define_string == nil and mw.text.trim(args[v] or "") ~= "" then _flag_use_user_define_string = true end
text = mw.text.trim(printer:printInfo(args[v] or "",v) or '')
if text ~= nil and text ~= '' then
body = body .. text ..'\n'
end
end
body = mw.text.trim(body)
if (_flag_use_user_define_string == true) and
(not (_flag_mass_output == true)) then body = body .. "[[Category:使用自定義Module:Number語句的頁面]]" end
return body
end
return p