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There has been a disconnect between our understanding of the universe's working at the micro and macro scale - that is, disagreement between quantum mechanics (QM) and general relativity (GR). A theory of vacuum-matter equilibrium is presented from scratch to bridge this gap. It is proposed that gravitons are both quanta of matter and gravitation, as every matter entity is believed to interact with gravitation. The vacuum has an energy density that gives rise to virtual-graviton pairs. Upon collision, a virtual-graviton may get stuck to the matter while a graviton from the matter gets ejected. It is possible as all gravitons are identical in mass and size. Equilibrium is thus established where matter erodes in the vacuum, and the vacuum condenses as matter. The probability of graviton exchange depends on the relative energies of the virtual-graviton and matter entity, as calculated in the inertial frame of reference (IFoR) decided by the state motion of vacuum energy at that place. If the rate of matter-vacuum equilibrium is taken as a constant, it leads to the notion of time and time dilation. The force appearing on a matter entity through collision with virtual-gravitons is calculated, and the expression is a hybrid of Newton's and Einstein's equation. It further helps to answer the queries related to phenomena like the flatness of galaxy rotation curves, misconceptions about relativistic mass and length contraction, the relation between time, gravity, and quantum entanglement (QE), and the composite nature of the universal gravitational constant.

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Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, it is written as $2^z = 3^kn + C$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by re-appearing, then the equation modifies to $2^z n =3^kn +C$. The article takes an elementary approach to this problem by stating that the inequality $2^z > 3^k$ must hold for $n$ to violate the Collatz conjecture. It leads to the inequality $z > 2k$ that helps obtain the relations $3^k/2^z = 3/4 - p$ and $2^z - 3^k = 2^z/4 + q$, where $p, q$ are some variables. The values of $p, q$ are determined by substitution in the $2^zn = 3^kn + C$, and the solution found is $(n, k, z, p, q) = (1, 1, 2, 0, 0)$

Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, the Collatz function is written as $f^k(n) = \frac{3^kn + C}{2^{z}}$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by reappearing as $2^in$, where $i \geq 1$, then the equation modifies to $n \left(1 - \frac{3^k}{2^{z}2^i}\right) = \frac{C}{2^{z}2^i}$. The article takes an elementary approach to this problem by calculating the bounds on the values of $\frac{C}{2^{z}2^i}$ and $1 - \frac{3^k}{2^{z}2^i}$. Correspondingly, an upper limit on the integer $n$ is placed that can re-appear in the sequence. The integer $n$ lies in the $(-\infty, 5)$ range, and the limit on the number of odd steps is $k < 3$. Finally, it is shown that no integer chain exists that does not lead to 1.