• Seven, the fourth prime number, is not only a Mersenne prime (since 2³ − 1 = 7) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a factorial prime, a lucky prime, a happy number (happy prime), a safe prime (the only Mersenne safe prime) and the fourth Heegner number.
• Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers. (See Lagrange's four-square theorem#Historical development.)
• Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree.
• n = 7 is the first natural number for which the next statement does not hold: "Two nilpotent endomorphisms from Cⁿ with the same minimal polynomial and the same rank are similar."
• 7 is the only dimension, besides the familiar 3, in which a vector cross product can be defined.
• 7 is the lowest dimension of a known exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere.
• 999,999 divided by 7 is exactly 142,857. Therefore, when a vulgar fraction with 7 in the denominator is converted to a decimal expansion, the result has the same six-digit repeating sequence after the decimal point, but the sequence can start with any of those six digits.^<1> For example, 1/7 = 0.142 857 142.... and 2/7 = 0.285 714 285...

In fact, if one sorts the digits in the number 142857 in ascending order, 124578, it is possible to know from which of the digits the decimal part of the number is going to begin with. The remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example, 628/7= 89 + 5/7; here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case, 628/7 = 89.714285. Another example, 5238/7 = 748 + 2/7, hence the remainder is 2, and this corresponds to number 2 in the sequence. In this case, 5238/7 = 748.285714.

• A seven-sided shape is a heptagon. The regular n-gons for n ≤ 6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons (including seven) are called heptagonal numbers. Seven is also a centered hexagonal number.
• Seven is the first integer reciprocal (multiplicative inverse) with infinitely repeating sexagesimal representation.
• There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.
• There are seven fundamental types of catastrophes.

Graph of the probability distribution of the sum of 2 six-sided dice

• When rolling two standard six-sided dice, seven has a 6 in 36 (or 1/6) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.
• The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved.

The night Max wore his wolf suit.. and made mischief of one kind.. and another.. his mother called him, "Wild thing!" and max said "I'll eat you up" so he was sent to bed without eating anything. That very night, in Max's room, a forest grew, and grew, and grew until his ceiling hung with vines and the walls became the world all around. And an ocean tumbled by with a private boat for Max. And he sailed off, through night and day, in and out of weeks, and almost over a year, to Where the Wild Things Are! And when he came to the place where the Wild Things are they roared their terrible roars and gnashed their terrible teeth and rolled their terrible eyes and showed their terrible claws until Max said "Be Still!" And tamed them with the magic trick of staring into all their yellow eyes without blinking once, and they were frightened, and called him the most wild thing of all and made him Kind of All Wild Things.
"And now", cried Max "Let the wild rumpus start!"

"Now stop!" Max said and sent the wild things off to bed without their supper. But Max, the King of All Wild Things, was lonely and wanted to be where someone loved him best of all. Then all around from far away across the world he smelled good things to eat, so he gave up being king of where the wild things are. But the wild things cried, "Oh, please don't go! We'll eat you up, we love you so!" But Max said "No!" and climbed into his boat and sailed off, through night and day, in and out of weeks, and almost over a year, into the night of his very own room where he found his supper waiting for him

.. and it was still hot.