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In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

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The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, 1 ( n ) = 2 n {\displaystyle \sigma _{1}(n)=2n} where 1 {\displaystyle \sigma _{1}} is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28.

It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefvre stated that Euclid's rule gives all perfect numbers,[17] thus implying that no odd perfect number exists. Euler stated: "Whether ... there are any odd perfect numbers is a most difficult question".[18] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.[19] All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.[20]

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.

A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.

I have been making your everyday meatballs (without the crumbs and cheese) and adding spaghetti for a long time; never thought baking the meatballs would make such a huge difference, but omg, it really makes them perfect! Even with an all-beef meatball, they were so soft and amazing! Thanks!

This truly is, for me, a perfect meatball recipe! Perfectly flavorful, moist, amazing texture. Next time I will definitely double the sauce, but other than that it was a home run! And also just as fast and easy as promised- I did the recipe plus bread and salad in around an hour and a half, which is fast for me :) Thanks!

Made this tonight. My new favorite spaghetti-meatballs recipe! How is something with so few ingredients, so good? Also, I am always going to use crushed tomatoes for sauce from now on! They are just the perfect texture. Thank you!

I made the meatballs last night. They were perfect. I only had beef and they were delicious, tender and cooked perfectly in 12 minutes. I always try new meatballs recipes, these are my favorite so far!

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I made my first Quiche for Thanksgiving. I had some pastry in the freezer from another recipe so I used used that for the crust. That worked great. After prebaking the crust I added a cup of grated parmesan to the bottom of the crust and then the filling, including 1.5 cups of cubed ham and a half cup of grated mozzarella. I had planned to add spinach but I forgot. I baked it in a Pyrex pie dish and everything fit perfectly. At the forty minute mark I covered the crust with aluminum foil to stop it getting too brown.

This recipes worked perfect for me but I stabbed the potato a lot more than recommended. I used a ninja air fryer and baking combo. Put some vegetarian impossible chili on top with mozzarella and green onions. So perfect.

tag_hash_108tag_hash_109tag_hash_110This flour blend is well, just about perfect! Substitute one cup of our Perfect Flour Blend for one cup of wheat flour in your favorite everyday recipes. Now you can make grandma's famous banana cake gluten free and no one will know!!

The adjective perfect emphasizes completeness, and also high quality and absence of defects or blemishes: a perfect diamond. Complete implies that a certain unit has all its parts or is fully developed or perfected, and may apply to a process or purpose carried to fulfillment: a complete explanation. Intact implies retaining completeness and original condition: a package delivered intact. Entire means whole, having unbroken unity: an entire book.

OAKLAND, Calif. -- Domingo German authored a perfect game against the Oakland Athletics on Wednesday night, as the New York Yankees starter allowed no hits and no walks on 99 pitches in an 11-0 victory.

The first pitcher born in the Dominican Republic to pitch a perfect game, German also became the first player in MLB history to reach that feat after allowing 10-plus runs in his previous start, according to ESPN Stats & Information research. He joins Don Larsen (1956), David Wells (1998) and David Cone (1999) as Yankees pitchers to achieve the milestone. Larsen's gem came in Game 5 of the 1956 World Series against the Brooklyn Dodgers.

"So exciting. When you think about something very unique in baseball, you know not many people have an opportunity to pitch a perfect game," German said. "To accomplish something like this in my career, you know, it's something that I am going to remember forever, be part of history, so exciting." ff782bc1db