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PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community. On February 13th 2013, PlanetMath.org was updated to use the new software system Planetary. Some release notes are here. Please report bugs in the Planetary Bugs Forum or on Github.

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Latest Messages  

[p] pseudoprimes in k(i) (contd) by akdevaraj 5:29 am
Although 561 is a Carmichael number in k(1) it is only a pseudoprime in k(i). If we have a composite number consisting of two primes of form 4m+3 and if it happens to be a pseudo to a base, say 2, it is also pseudo to the base (number + i). Example: 341 = 11*31; this is pseudo to base 2. It is also pseudo to base (341 + i) and base ( 341 + 2i).

[p] pseudoprimes in k(i) by akdevaraj Jul 29
341 is a pseudoprime to base 2 (which is in k(1). It is also a pseudoprime to base (341 + i). Hence it is a pseudoprime in k(i) also. Similarly 561, a Carmichael number is a pseudoprime in k(i) as it is pseudo to the base (187 + i) as well as the base (561 + i). Interestingly it is also pseudo to the base (561 + 2i).

Some formulas of Arithmetic progression/series by burgess Jul 29
An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Example: 2,4,6,8,10….. Arithmetic Series : The sum of the numbers in a finite arithmetic progression is called as Arithmetic series. Example: 2+4+6+8+10….. nth term in the finite arithmetic series Suppose Arithmetic Series a1+a2+a3+…..an Then nth term an=a1+(n-1)d Where a1- First number of the series an- Nth Term of the series n- Total number of terms in the series d- Difference between two successive numbers Sum of the total numbers of the arithmetic series Sn=n/2*(2*a1+(n-1)*d) Where Sn – Sum of the total numbers of the series a1- First number of the series n- Total number of terms in the series d- Difference between two successive numbers Example: Find n and sum of the numbers in the following series 3 + 6 + 9 + 12 + x? Here a1=3, d=6-3=3, n=5 x= a1+(n-1)d = 3+(5-1)3 = 15 Sn=n/2*(2a1+(n-1)*d) Sn=5/2*(2*3+(5-1)3)=5/2*18 = 45 I hope the above formulae are helpful to solve your math problems>

Procedure to calculate the day of the week for a given date. by burgess Jul 28
To calculate the day of the week for a given date, first of all we need to find out the number of odd days. Today I thought of sharing a beautiful problem I learned in my school, though it is easy, it is tricky too. Odd Days are number of days more than the complete number of weeks in given period. Leap Year is the year which is divisible by 4. A normal year has 365 days A leap year has 366 days One normal year = 365 days = 52weeks + 1day One normal year has one odd day One leap year = 366 days = 52weeks + 2days One leap year has two odd days 100 years = 76 ordinary years + 24 leap years = 5200 weeks + 124 days = 5217 weeks + 5 days 100 years have 5 odd days 400 years have (20+1) 0 odd days The number of odd days and the corresponding day of the week is given below 0-Sunday 1-Monday 2-Tuesday 3-Wednesday 4-Thursday 5-Friday 6-Saturday So by finding out the number of odd days you can find out the day of the week. I hope this procedure Will be helpful in solving math problems in exams. Thanks. >

[p] Fermat's theorem in k(i) (c0ntd) by akdevaraj Jul 21
This base, however , does not work in the case of primes having shape 4m+3. A base that works is 1 + i. Example ((1+i)^102 + I)/103 = -21862134113449i.

[p] Fermat's theorem in k(i) (c0ntd) by akdevaraj Jul 20
What is the nature of a, the base? When p has the shape 4m+1 a has the shape of a prime factor of a number having the same shape. Example: Let p = 61. Then ((4 + i)^60 - 1 )/61 = -71525089284120116591639000327021600 + 11369162311133702688684197835211600i

[p] Fermat's theorem in k(i) (c0ntd) by akdevaraj Jul 19
Before giving some further generalisations let me give some examples: case a) ((1+I)^30 + I)/31 = -1057i. ((1 + i )^102 + i)/103 = -21862134113449i Case b) ((1 + i)^12 - 1)/13 = -5. (( 1 + i ) ^100 - 1)/101 = -11147523830125

[p] Fermat's theorem in k(i) (c0ntd) by akdevaraj Jul 17
Although Hardy and Wright have formulated the above theorem in their book ("An introduction to he theory of numbers " we can see how it works with the aid of software like pari. The four examples illustrate this. Now for a few genralisations: a) If p is a prime of form 4m+3, then ((a^(p-1)+ I)/p is congruent to 0 (mod(p)). b) If p is a prime of form 4m+1, then ((a^(p-1) - 1)/p is congruent to 0 (mod(p)).

[p] Fermat's theorem in k(i) by akdevaraj Jul 14
There are four unities in k(i) viz 1, -1, i and -i. Four examples are given here to illustrate Fermat's theorem in k(i). a)((2+3i)^2-1)/3 = -2 +4i b) ((3+2i)^2 + 1)/3 = 2 + 4i c) ((10 + i)^2 + i)/3 = 33 + 7i and d) ((14 +i)^2 - i)/3 = 65 + 9i.

how to determinate the unknowns of this simplex tableau by anouarattn Jul 11
Considering 2 simplex tableau encountered when solving a linear program http://i62.tinypic.com/9r80ic.jpg determine the value of each of the following items(unknowns) : p q r that appear in tables http://i61.tinypic.com/mcwx9u.jpg And please if someone has another example like this one please give it to me>

[P] I just read your article and by Happy19th Jun 30
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[p] A good question by pahio Jun 25
Please see "sophomore's dream" in Wikipedia.

[p] easy exercice by Ron Castillo Jun 23
1.- Is False 2.- Is True 1.- Is False 1.- Is true, iff b is rational. Regards, Ronald.

[P] failure functions - another exampleL by akdevaraj Jun 23
Let our definition of a failure be a composite number which is also a multiple of 11. Let the parent function be 2^n + 7 (n belongs to N ). Then n = 2 + Eulerphi(11) is a failure function. Also n = 2^(1 + Eulerphi(Eulerphi(11)) is also a failure function.