Lagranges four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal. A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals.The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.A solid figure with four faces as well as four vertices is a tetrahedron, which is the smallest possible number of faces and vertices a polyhedron can have. The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid. It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron. Tetrahedra can be inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb.
Alterations in blood flow: immobilization (after surgery, long-haul flight), injury, pregnancy (also procoagulant), obesity (also procoagulant), cancer (also procoagulant)
Factors in the vessel wall: surgery, catheterizations causing direct injury ("endothelial injury")
Factors affecting the properties of the blood (procoagulant state):
Estrogen-containing medication (transgender hormone therapy, menopausal hormone therapy and hormonal contraceptives)
Genetic thrombophilia (factor V Leiden, prothrombin mutation G20210A, protein C deficiency, protein S deficiency, antithrombin deficiency, hyperhomocysteinemia and plasminogen/fibrinolysis disorders)
Acquired thrombophilia (antiphospholipid syndrome, nephrotic syndrome, paroxysmal nocturnal hemoglobinuria)
Cancer (due to secretion of pro-coagulants)Although most pulmonary embolisms are the result of proximal leg deep vein thrombosis (DVTs), there are still many other risk factors that can also result in a pulmonary embolism. Risk factors include:
Varicose veins caused by vascular damage
Pulmonary hypertension
Diabetes
Traumatic hip fractures that immobilize the patient
Joint fixation (primarily in the legs)

Underlying causes
After a first PE, the search for secondary causes is usually brief. Only when a second PE occurs, and especially when this happens while still under anticoagulant therapy, a further search for underlying conditions is undertaken. This will include testing ("thrombophilia screen") for Factor V Leiden mutation, antiphospholipid antibodies, protein C and S and antithrombin levels, and later prothrombin mutation, MTHFR mutation, Factor VIII concentration and rarer inherited coagulation abnormalities. Diagnosis
To diagnose a pulmonary embolism, a review of clinical criteria to determine the need for testing is recommended.