## Posts filled under #catphoto

nimo de segunda... tima semana amigos!! Mordidinhas do Wasabi #cats #catstagram #lovecats #catsofinstagram #pets #instacats #catlover #gatos #gatosdeinstagram #neko #filhode4patas #bichanosbr #meuamigogatooficial #ronron #insta_meows #instagatos #euamogatos #instapets #catphoto #catlovers #ilovemycat #amorfelino #maedegato #petstagram #instagrampets #gateira #gatowasabi ************************************************* Follow my friends / Sigam meus amigos: @aslam.e.maya @irina_edith @gagaemia @gatolatifundiario @billyboy5190 @polpologato @mingoemell @luigi_lali_raja @molequeecia *************************************************

## An extract on #catphoto

Euclid proved that 2p1(2p 1) is an even perfect number whenever 2p 1 is prime (Euclid, Prop. IX.36). For example, the first four perfect numbers are generated by the formula 2p1(2p 1), with p a prime number, as follows: for p = 2: 21(22 1) = 6 for p = 3: 22(23 1) = 28 for p = 5: 24(25 1) = 496 for p = 7: 26(27 1) = 8128. Prime numbers of the form 2p 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p 1 with a prime p are prime; for example, 211 1 = 2047 = 23 89 is not a prime number. In fact, Mersenne primes are very rareof the 2,270,720 prime numbers p up to 37,156,667, 2p 1 is prime for only 45 of them. Nicomachus (60-120 AD) conjectured that every perfect number is of the form 2p1(2p 1) where 2p 1 is prime. Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of that form. It was not until the 18th century that Leonhard Euler proved that the formula 2p1(2p 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the EuclidEuler theorem. As of January 2016, 49 Mersenne primes are known, and therefore 49 even perfect numbers (the largest of which is 274207280 (274207281 1) with 44,677,235 digits). An exhaustive search by the GIMPS distributed computing project has shown that the first 45 even perfect numbers are 2p1(2p 1) for p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, and 37156667 (sequence A000043 in the OEIS). Four higher perfect numbers have also been discovered, namely those for which p = 42643801, 43112609, 57885161, and 74207281, though there may be others within this range. It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. As well as having the form 2p1(2p 1), each even perfect number is the (2p 1)th triangular number (and hence equal to the sum of the integers from 1 to 2p 1) and the 2p1th hexagonal number. Furthermore, each even perfect number except for 6 is the ((2p + 1)/3)th centered nonagonal number and is equal to the sum of the first 2(p1)/2 odd cubes: 6 = 2 1 ( 2 2 1 ) = 1 + 2 + 3 , 28 = 2 2 ( 2 3 1 ) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 1 3 + 3 3 , 496 = 2 4 ( 2 5 1 ) = 1 + 2 + 3 + + 29 + 30 + 31 = 1 3 + 3 3 + 5 3 + 7 3 , 8128 = 2 6 ( 2 7 1 ) = 1 + 2 + 3 + + 125 + 126 + 127 = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3 , 33550336 = 2 12 ( 2 13 1 ) = 1 + 2 + 3 + + 8189 + 8190 + 8191 = 1 3 + 3 3 + 5 3 + + 123 3 + 125 3 + 127 3 . {\displaystyle {\begin{aligned}6&=2^{1}(2^{2}-1)&&=1+2+3,\\[8pt]28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7=1^{3}+3^{3},\\[8pt]496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3},\\[8pt]8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3},\\[8pt]33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}.\end{aligned}}} Even perfect numbers (except 6) are of the form T 2 p 1 = 1 + ( 2 p 2 ) ( 2 p + 1 ) 2 = 1 + 9 T ( 2 p 2 ) / 3 {\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}} with each resulting triangular number (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with 3, 55, 903, 3727815, .... This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2p1(2p 1) with odd prime p and, in fact, with all numbers of the form 2m1(2m 1) for odd integer (not necessarily prime) m. Owing to their form, 2p1(2p 1), every even perfect number is represented in binary as p ones followed by p 1 zeros: 610 = 1102 2810 = 111002 49610 = 1111100002 812810 = 11111110000002 3355033610 = 11111111111110000000000002. Thus every even perfect number is a pernicious number. Note that every even perfect number is also a practical number (c.f. Related concepts).