So if it's not known whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol and compare it with Euler's formula ; if they differ modulo n, then n is composite ; if they're the same modulo n for many different values of a, then n is " probably prime ".

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

So, the prime number theorem can also be written as π ( x ) ~ Li ( x ).

So, just like one can find the value of an Integral ( f = dF ) over a 1-dimensional manifolds () by considering the anti-derivative ( F ) at the 0-dimensional boundaries (), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals ( dω ) over n-dimensional manifolds ( Ω ) by considering the anti-derivative ( ω ) at the ( n-1 )- dimensional boundaries ( dΩ ) of the manifold.

So the fundamental theorem reads:

So if there is such a predicate P and a theorem:

So construct the midpoint H between O and P, and draw a circle centered at H through O and P. By Thales ' theorem, the sought point T is the intersection of this circle with the given circle k, because that is the point on k that completes a right triangle OTP.

So the Atiyah – Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.

So the index theorem can be proved by checking it on these particularly simple cases.

So, a key criticism is that the theorem is almost always inapplicable in economic reality, because real-world transaction costs are rarely low enough to allow for efficient bargaining.

So, the fact that the time-reversed version of the fluctuation theorem does not ordinarily give accurate predictions in the real world is reason to think that the nonequilibrium state of the universe at the present moment is not simply a result of a random fluctuation, and that there must be some other explanation such as the Big Bang starting the universe off in a low-entropy state ( see below ).

So far it has won the " world cup for theorem provers " ( the CADE ATP System Competition ) in the most prestigious CNF ( MIX ) division eleven times ( 1999, 2001 – 2010 ).

Then all the lines which are common tangents to both P = 0 and Q = 0 are tangent to C. So, by the AF + BG theorem, the tangential equation of C has the form HP + KQ = 0.

So, the truth of the Agoh-Giuga conjecture combined with Wilson's theorem would give: a number p is prime iff

So, it ’ s very much possible that Pythagoras learned this theorem during his visit to India but hid his source of knowledge he went back to Greece!

So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so pqr = qpr, which proves the dual of Pappus's theorem.

So, the Weinberg – Witten theorem applies and we can't get a massless spin-2 ( i. e. helicity ± 2 ) composite / emergent graviton.

So, we can have spin-3 / 2 and spin-2 bound states with arbitrarily small masses but still not violate the theorem.

So to generate a name for the universal distinctive of chairs, take the predicate " chair " and add " ness ", which yields " chairness ".

So, < τέλειος > (< téleios >, meaning " perfect " and " complete ") yields < τέλεια > (< téleia >, " perfectly ") and < τελείως > (< teleíos >, " completely ").

So the " over-reinforced concrete " beam fails by crushing of the compressive-zone concrete and before the tension zone steel yields, which does not provide any warning before failure as the failure is instantaneous.

So, future food-production increases will have to come from higher yields.

So the net acceleration ( actually a deceleration ) of the rotation of the Earth, or the change in the length of the mean solar day ( LOD ), is + 1. 7 ms / day / cy.

So in May 1893 a new regulation to all chiefs of police, stated that the police should not intervene, if the two last fields in the flag were longer than 6 / 4 as long as these did not exceed 7 / 4, and provided that this was the only rule violated.

* 1974 – " Psychedelic Warlords " / " It's So Easy " ( 7 " single )

So a strand will never be 7 mm, but will be 6. 5 – 7 mm.

So, a diver with a 3 litre oxygen cylinder filled to 200 bar who leaves 25 % in reserve will be able to do a 450 minute = 7. 5 hour dive ( 3 L × 200 bar × 0. 75 / 1 ).

So, to completely react with 85. 0 g of ( 0. 532 mol ), 28. 7 g ( 1. 06 mol ) of aluminium are needed.

So the mixture is 17. 8, 1. 6, 5. 5 compared to Chloroform at 17. 8, 3. 1, 5. 7.

So, again with the aim of probing the Spanish defenses of that city, on March 7, 1740 Vernon left Port Royal in command of two fire-ships, three bombards and a steamer, reaching waters of Cartagena on March 13.

On August 7, 1973 Norman entered AIR studios in London to record his favorite album, the second album in his Trilogy, So Long Ago the Garden, which was produced again by Edwards, Hand, and Miller.

* Lauren Froderman-Winner of So You Think You Can Dance ( Season 7 )

* Kent Boyd, runner-up on Season 7 ( 2010 ) of So You Think You Can Dance.

* " September's Not So Far Away " ( 7 ", Sarah 044, 1991 )

Jack Teagarden's compositions included " I've Got ' It '" with David Rose, " Shake Your Hips ", " Big T Jump ", " Swingin ' on the Teagarden Gate ", " Blues After Hours ", " A Jam Session at Victor ", " It's So Good ", " Pickin ' For Patsy " with Allan Reuss, " Texas Tea Party " with Benny Goodman, " I'm Gonna Stomp Mr. Henry Lee " with Eddie Condon, " Big T Blues ", " Dirty Dog ", " Makin ' Friends " with Jimmy McPartland, " That's a Serious Thing ", and " Jack-Armstrong Blues " with Louis Armstrong, recorded on December 7, 1944 with the V-Disc All-Stars and released as V-Disc 384A.

So all prime numbers are of the form 30k + i for i = 1, 7, 11, 13, 17, 19, 23, 29 ( i. e. for i < 30 such that gcd ( i, 30 ) = 1 ).

So on September 28, 1981, Morning dropped the days of the week from its title ( except for Sunday Morning ), and was extended to 90 minutes and added Diane Sawyer as co-host ; in the process, Captain Kangaroo was reduced to a half-hour daily and pushed to an earlier time period ( 7: 00 a. m .).

* So You Wanna Go Back to Egypt ( May 7, 1980 )

* So, Mr Bremer, where did all the money go ?, The Guardian, July 7, 2005

On August 7, 2008, Pink's single " So What " was leaked online, and radio stations across Australia were quick to give it massive airplay.

So a quasar at an absolute magnitude of − 25. 5 is 100 times brighter than our galaxy ( because ( 10 < sup > 0. 4 </ sup >)< sup >(− 20. 5 -(− 25. 5 ))</ sup >

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

So, substituting the angular momentum and moment of inertia into E < sub > rot </ sub >, the rotational energy levels of a diatomic molecule are given by:

So P < sub > n </ sub > agree with the Bernoulli polynomials on the interval ( 0, 1 ) and are periodic with period 1.

So the parameters are: a — distance from center C to either vertex b — length of a perpendicular segment from each vertex to the asymptotes c — distance from center C to either Focus point, F < sub > 1 </ sub > and F < sub > 2 </ sub >, and θ — angle formed by each asymptote with the transverse axis.

So the logarithm function is in fact a group isomorphism from the group ( R < sup >+</ sup >,< big >×</ big >) to the group ( R ,< big >+</ big >).

) So far, observations show any rotation of the Universe is very slow ( no faster than once every 60 · 10 < sup > 12 </ sup > years ( 10 < sup >− 13 </ sup > rad / yr )), and debate persists over whether there is any rotation.

So in the example above, the has simply been replaced by < code >< nowiki > http :// usemod. com / cgi-bin / mb. pl ?</ nowiki ></ code > in creating the target of the HTML rendered link.

So if there are m factors, there will be 2 < sup > m </ sup > idempotents.

So, in these cases, the lethal concentration may be given simply as LC < sub > 50 </ sub > and qualified by a duration of exposure ( e. g., 10 minutes ).

So < sup > 256 </ sup > Fm was produced by the decays of cyclotron-synthesized mendelevium.

So if L ( r ) is proportional to r < sup >− 2 </ sup >, then for to be finite, N ( r ) must be proportional to r < sup > b </ sup >, where b < 1.

So beginning at the left and moving to the right, find the first ( and in this case, the only ) parenthesis, that is, ( 2 + 2 < sup > 2 </ sup >).

So the correct value for our original expression, 4 × 2 < sup > 2 </ sup > − ( 2 + 2 < sup > 2 </ sup >), is 10.

So the output contains sinusoidal terms with frequencies at the sum ω < sub > 1 </ sub > + ω < sub > 2 </ sub > and difference ω < sub > 1 </ sub >-ω < sub > 2 </ sub > of the two original frequencies.