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Question Number 159461 Answers: 0 Comments: 4

≺PRIME-BIRTHDAYS≻ Do you know ′Prime1611′?... No,no it′s not an ID of the forum- member.It is a person who was born on November 16, 0001.On his birthday astrologers formed a string from his birthdate in the way: ′ddmmyyyy′: ′16110001′ The astro- logers observed that number containing in the string: 16110001 is a prime number.Thus they gave him name ′Prime1611′ They also suggested that Mr Prime1611 should celebrate his birthday only when the number ddmmyyyy be prime number.Recently He celebrated his birthday on 16-11-2021 as the number 16112021 is prime. How many birthdays has he celebrated upto his recent birthday when he had celebrated his first birthday on 16-11-0001? You may use calculator also. Question connected with Q#159421

$$\:\:\:\:\:\:\:\:\:\:\prec\mathbb{PRIME}-\mathcal{BIRTHDAYS}\succ \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:'\mathrm{Prime1611}'?... \\ $$$$\mathrm{No},\mathrm{no}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{ID}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}- \\ $$$$\mathrm{member}.\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{person}\:\mathrm{who}\:\mathrm{was}\:\mathrm{born} \\ $$$$\mathrm{on}\:\mathrm{November}\:\mathrm{16},\:\mathrm{0001}.\mathrm{On}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{astrologers}\:\mathrm{formed}\:\mathrm{a}\:\mathrm{string} \\ $$$$\mathrm{from}\:\mathrm{his}\:\mathrm{birthdate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{way}: \\ $$$$'\mathrm{ddmmyyyy}':\:'\mathrm{16110001}'\:\mathrm{The}\:\mathrm{astro}- \\ $$$$\mathrm{logers}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{number}\:\mathrm{containing} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{string}:\:\mathrm{16110001}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime} \\ $$$$\mathrm{number}.\mathrm{Thus}\:\mathrm{they}\:\mathrm{gave}\:\mathrm{him}\:\mathrm{name} \\ $$$$'\mathrm{Prime1611}'\:\mathrm{They}\:\mathrm{also}\:\mathrm{suggested}\:\mathrm{that} \\ $$$$\mathrm{Mr}\:\mathrm{Prime1611}\:\mathrm{should}\:\mathrm{celebrate}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{ddmmyyyy}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{number}.\mathrm{Recently} \\ $$$$\mathrm{He}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{2021}\:\mathrm{as}\:\mathrm{the}\:\mathrm{number}\:\mathrm{16112021} \\ $$$$\mathrm{is}\:\mathrm{prime}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{birthdays}\:\mathrm{has}\:\mathrm{he}\:\mathrm{celebrated} \\ $$$$\mathrm{upto}\:\mathrm{his}\:\mathrm{recent}\:\mathrm{birthday}\:\mathrm{when}\:\mathrm{he} \\ $$$$\mathrm{had}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{first}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{0001}? \\ $$$$\mathrm{You}\:\mathrm{may}\:\mathrm{use}\:\mathrm{calculator}\:\mathrm{also}. \\ $$$$\mathrm{Question}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{Q}#\mathrm{159421} \\ $$

Question Number 158914 Answers: 0 Comments: 0

Σ_(n=0) ^∞ (1/((3n+1)^3 ))=?

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$

Question Number 158270 Answers: 0 Comments: 0

82,1336,18670,240004,2933338,34666672,400000006,? is there a valid pattern for these numbers?

$$\mathrm{82},\mathrm{1336},\mathrm{18670},\mathrm{240004},\mathrm{2933338},\mathrm{34666672},\mathrm{400000006},? \\ $$$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{valid}\:\mathrm{pattern}\:\mathrm{for}\:\mathrm{these}\:\mathrm{numbers}? \\ $$

Question Number 157853 Answers: 0 Comments: 1

Question Number 157219 Answers: 0 Comments: 0

Σ_(0<n) (((−1)^(n−1) n)/(sinh(πn)))=(1/(4π)) prove

$$\underset{\mathrm{0}<\boldsymbol{\mathrm{n}}} {\sum}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \boldsymbol{\mathrm{n}}}{\boldsymbol{\mathrm{sinh}}\left(\pi\boldsymbol{\mathrm{n}}\right)}=\frac{\mathrm{1}}{\mathrm{4}\pi}\:\:\:\:{prove} \\ $$

Question Number 156419 Answers: 1 Comments: 1

x is positive integer number can you check if Q=((((x+2)^4 −x^4 ))^(1/3) is a natural number

$$\mathrm{x}\:\mathrm{is}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\mathrm{can}\:\mathrm{you}\:\mathrm{check}\:\mathrm{if}\:\mathrm{Q}=\sqrt[{\mathrm{3}}]{\left(\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{4}} −\mathrm{x}^{\mathrm{4}} \right.} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number} \\ $$

Question Number 156126 Answers: 1 Comments: 1

cos(π/5)=...? with solution pls

$$\:\:\:\mathrm{cos}\frac{\pi}{\mathrm{5}}=...?\:\:\mathrm{with}\:\mathrm{solution}\:\mathrm{pls} \\ $$

Question Number 156201 Answers: 0 Comments: 1

A=[((x^n ((x^n^2 ((x^n^3 ∙∙∙∙(x^n^n )^(1/n) ))^(1/n) ))^(1/n) ))^(1/n) ]^(1/n)

$$\:\:{A}=\left[\sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}} \sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}^{\mathrm{2}} } \sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}^{\mathrm{3}} } \centerdot\centerdot\centerdot\centerdot\sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}^{\mathrm{n}} } }}}}\right]^{\frac{\mathrm{1}}{\mathrm{n}}} \\ $$

Question Number 155571 Answers: 2 Comments: 0

Find the cube root of one .Hence show that the sum of the root is equal to zero

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\mathrm{one}\:.\mathrm{Hence} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$

Question Number 155133 Answers: 2 Comments: 0

Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^3 ))=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$

Question Number 154672 Answers: 1 Comments: 0

how many positive x≤10 000 integers are such that 2^x −x^2 is divisible by 7?

$$\: \\ $$$$\:\mathrm{how}\:\mathrm{many}\:\mathrm{positive}\:{x}\leqslant\mathrm{10}\:\mathrm{000}\:\mathrm{integers}\:\mathrm{are}\:\: \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{2}^{{x}} −{x}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}? \\ $$$$\: \\ $$

Question Number 153916 Answers: 0 Comments: 0

The value of Σ_(n=0) ^∞ (((3_n )(2_n )x^n )/((1_n )n!)) β(2,n+1) is a. (1/2)Σ_(n=0) ^∞ (2_n )(x^n /(n!)) b. (1/2)Σ_(n=0) ^∞ (((3_n )(2_n ))/((1_n ))) (x^n /(n!)) c. (1/2)Σ_(n=0) ^∞ (((2_n )x^n )/((1_n )n!)) d. (1/3)Σ_(n=0) ^∞ (((3_n )x^n )/((1_n )n!))

$${The}\:{value}\:{of}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!}\:\beta\left(\mathrm{2},{n}+\mathrm{1}\right)\:{is} \\ $$$${a}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\mathrm{2}_{{n}} \right)\frac{{x}^{{n}} }{{n}!} \\ $$$${b}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right)}{\left(\mathrm{1}_{{n}} \right)}\:\frac{{x}^{{n}} }{{n}!} \\ $$$${c}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!} \\ $$$${d}.\:\frac{\mathrm{1}}{\mathrm{3}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{3}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!} \\ $$

Question Number 153458 Answers: 0 Comments: 1

Given a set consisting of 22 integer A={±a_1 ,±a_2 ,...,±a_(11) }. Show that exist subset of S with properties (1) for every i=1,2,3,...,11 have least one between a_i or −a_i element of S (2)the sum all possible numbers in S divisible by 2015

$${Given}\:{a}\:{set}\:{consisting}\:{of}\:\mathrm{22}\:{integer} \\ $$$$\:{A}=\left\{\pm{a}_{\mathrm{1}} ,\pm{a}_{\mathrm{2}} ,...,\pm{a}_{\mathrm{11}} \right\}.\:{Show}\:{that} \\ $$$${exist}\:{subset}\:{of}\:{S}\:{with}\:{properties} \\ $$$$\left(\mathrm{1}\right)\:{for}\:{every}\:{i}=\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{11}\: \\ $$$$\:{have}\:{least}\:{one}\:{between}\:{a}_{{i}} \:{or}\:−{a}_{{i}} \\ $$$$\:{element}\:{of}\:{S} \\ $$$$\left(\mathrm{2}\right){the}\:{sum}\:{all}\:{possible}\:{numbers} \\ $$$${in}\:{S}\:{divisible}\:{by}\:\mathrm{2015} \\ $$

Question Number 151265 Answers: 1 Comments: 3