Representation of a number we write.

I think that the only way to record the desired polynomial is to use the solutions of any equation.

Knowing the solutions of this equation and substituting them into the linear Diophantine equation.

variables which are solutions of this equation. Then the solution of the first equation can be written as.]]>]]>

]]>

https://brilliant.org/wiki/converse-of-intermediate-value-theorem/

]]>A=2x3x5x7x11x13x17........ up to

m=Any multiple

A-pm= A number factorable by a factor of p

if p has one.To get A-pm, go through the primes, subtracting p as many times as you like.

Example:

p=129

A=2x3x5x7x11

2x3x5x7=210 210-129=81

81x11=891 891-(129x6)=117

A-129m=117

117 will have a factor the same as p if it has any.

117/129=39/43 129 is NOT prime (Common denominator =3)

In this way we can find out if p is composite without ever having to use a number

. Might be useful for computers.]]>Example:

p=130

Rd. Up to nearest prime= 13

Next prime after that = 17

17-1=16

Largest prime gap <130 = 16 (Correct)

This works because the greatest number of composites between two primes occurs when factors are not combined. So what could have been two composites is actually just one, like 15=3x5. To create the greatest possible number of composites I start at 2 not 0. 0 has an infinite number of prime factors, and so the greatest gap between the next repeat will occur after 0. Starting with the smallest composite which is NOT combined factors, I move up. Deleting all numbers factorable by primes less than the square root. The first time I attain TWO primes is when I reach the second prime after the square root. So this -1 is the gap required to create two non-composites with greatest possible occurrence of composites.

Largest prime gap <p =

Rd. Up to second nearest prime -1.]]>The formula is too complicated to remember. It can be remembered much more easily by Implementing Faulhaber's formula. It says,

1ˣ +2ˣ +3ˣ +4ˣ ......nˣ =[1/(1+x)](aB₀nˣ⁺¹+bB₁nˣ+cB₂nˣ⁻¹............yBₓn), where Bₙ is the nth Bernoulli no. and a,b,c,.....y are the consecutive terms of (x+1)th row of Pascal's Triangle.

For e.g.

1¹¹+2¹¹+3¹¹+4¹¹.....7¹¹=(1/12)(1B₀n¹²+12B₁n¹¹+66B₂n¹⁰+220B₃n⁹+495B₄n⁸+792B₅n⁷+924B₆n⁶+792B₇n⁵+495B₈n⁴+220B₉n³+66B₁₀n²+12B₁₁n)

(1+B)ⁿ⁺¹-Bₙ₊₁=0

For e.g.

for,B₁ n=1 so

(1+B)²-B₂=0

⇒1+B₂+2B-B₂=0

⇒1=2B=0

⇒B=-0.5

However in this ( the formula for sums of powers )formula B₂=+0.5]]>

Sorry, haven't been keeping up with the "this is cool" feed. But nice job!

]]>http://www.mathisfunforum.com/viewtopic.php?id=20868

The most amazing things about it, were all personal. I became further convinced that EM was the only way to go if one desired to get the right answer an infuriating amount of times. Also, we were 2 of only 4 people in the world who knew that those Stanford and Duke university guys were not arguing with Marilyn. That would have been difficult enough, they were arguing with their own methods and colleagues who had hashed this problem to death in a statistics journal, many years before. RIPOSTP.

]]>URL:http://www.mathsisfun.com/numbers/infinity.html

]]>I am a guy who came into this forum in 2009. I currently live in the badlands of Florida.

did you become administrator

In March of 1990 at the tender age of 70 I decided to devote most of my remaining brain cells to mathematics. This was primarily the result of a dare by people who believed that I was a loser and could not stick to anything for more than a short time. They were partially right but I could stick to something if I wanted to. So, in that month I decided that I would teach myself mathematics and computers full time. I also made the promise that at the end of 20 years if I could not do any real math, I would quit forever. In April of 2009 ( the 19th year ) I decided it was time to see if I had succeeded.

I noticed the MathsisFun forum and one character in particular, the legendary JaneFairfax. She seemed to be the best problem solver on the forum and naturally I wanted to see how I would do against the best they had.

I did not do anything that I considered amazing for the first couple of weeks until the tongzilla problem appeared. When I solved that one, I knew I had succeeded. I had learned mathematics using my own methods and my own ideas. I had discovered EM. Naturally, my original detractors suddenly forgot their accusations but I did not.

Then I had to calm my mind so that I would not forget that what I had done was not amazing, it was not even difficult. To do that I used the techniques that pappym taught me from your country. They were much harder than mathematics and I am still working on them.

Why am I an administrator? Beats me...only MIF can answer that question. But it has been a privilege to serve here and to know the people that are here. A privilege I do not deserve but have been given to me anyway.

]]>1,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192..........

which is nothing but powers of 2.So,it is actually

2^0,2^0,2^1,2^2,2^3,2^4,2^5,2^6.........

There is also a formula first noticed by Leonhard Euler which proves that the set of primes is endless. The formula is

x²+x+41 is always a prime no. if x is an integer.

This cannot be true: just take x = 41 for instance, which clearly factorises into non-trivial factors. Euler found that this polynomial produces 40 distinct primes for the first 40 values.

In fact, it can be shown that such a polynomial cannot exist.

]]>p=35

Ps=77777777777777777777777777777777813