If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a conjecture, as opposed to an ultimate goal: a theorem that has been proved.

If, for some n, such an i doesn't exist, we say that n has infinite total stopping time and the conjecture is false.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1.

If the strong Goldbach conjecture is true, the weak Goldbach conjecture will be true by implication.

This precept, from one of Bacchylides ' extant fragments, was considered by his modern editor, Richard Claverhouse Jebb, to be typical of the poet's temperament: " If the utterances scattered throughout the poems warrant a conjecture, Bacchylides was of placid temper ; amiably tolerant ; satisfied with a modest lot ; not free from some tinge of that pensive melancholy which was peculiarly Ionian ; but with good sense ..."

If Rivers ever did come to access the veiled memory then he does not appear to make a note of it so the nature of the experience is open to conjecture.

The conjecture is stated in terms of three positive integers, a, b and c ( whence comes the name ), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d cannot be much smaller than c.

If G, H are graphs, we define a map φ: E ( G ) —> E ( H ) to be cycle-continuous if the pre-image of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every bridgeless graph has a cycle-continuous mapping to the Petersen graph.

Haupt's critical work is distinguished by a combination of the most painstaking investigation with bold conjecture ; his oft-cited dictum that " If the sense requires it, I am prepared to write Constantinopolitanus where the MSS have the monosyllabic interjection o " well expresses this boldness.

If the Hodge conjecture holds for Hodge classes of degree p, p < n, then the Hodge conjecture holds for Hodge classes of degree 2n − p.

If this conjecture is correct, quipus are the only known example of a complex language recorded in a 3-D system.

If the conjecture is false, a counterexample would take the form of a graph with minimum degree three having no power-of-two cycles.

If π ( x ) is the number of primes up to and including x then the conjecture states that

If the first Hardy – Littlewood conjecture holds, then the first such k-tuple is expected for x greater than 1. 5 × 10 < sup > 174 </ sup > but less than 2. 2 × 10 < sup > 1198 </ sup >.

" Despite its simplicity, it suggests a deeper conjecture: " If is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?

If a lower bound ( for the function value ) could be found for every one of these configurations that was greater than the value of the function for the cubic close packing arrangement, then the Kepler conjecture would be proved.

If the Birch and Swinnerton-Dyer conjecture was proved, Tunnell's theorem would give a full resolution to the congruent number problem.

If the conjecture is true, this would imply the nonexistence of odd perfect numbers.

If the unique games conjecture is true, then it is impossible to approximate MAX 2-SAT, balanced or not, with an approximation constant better than 0. 943 ... in polynomial time.

In mathematics, the Sato – Tate conjecture is a statistical statement about the family of elliptic curves E < sub > p </ sub > over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If N < sub > p </ sub > denotes the number of points on E < sub > p </ sub > and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for N < sub > p </ sub >.

If you were a man '' --

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If someone were to drop a match in here, this place would go up like a haystack ''.

If the other pilots were worried, they did not show it.

If it were the enemy, tactically his position was correct.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

If an automobile were approaching him, he would know what was required of him, even though he might not be able to act quickly enough.

If there were only darkness, all would be clear.

If there were only the mess, all would be clear ; ;

If these people were denied a voice ( do they have a moral right to a voice??

If we were creating a wholly new society, we could insist that our social, political, economic and philosophic institutions foster rather than hamper man ; ;

If living Jews were unavailable for study, the Bible was at hand.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If she were not at home, Mama would see to it that a fresh white rose was there.

If the Stearns were not there, grace would be omitted.

If anything, the conservative Democrats were more opposed to Hearst than the Republicans.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.

If there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the Hessian troops sent here willy-nilly by the Hessian Government to fight for England in the 1770's were mercenaries, what shall we call the UN troops sent to the Congo willy-nilly by their governments to fight for the United Nations??

If the UN troops are not mercenaries then the Hessians were not mercenaries either.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.