Mathematical conjecture is based on certain mathematical facts and contains valuable imagination elements based on mathematical facts; without mathematical facts as a basis, propositions that can be obtained by random guessing cannot be called " Mathematical Conjecture”. Mathematical conjectures are usually put forward using analogy and induction methods, or they emerge from inspiration or intuition. For example, the Chinese mathematician and linguist Zhou Haizhong, based on the known Mersenne prime numbers and their arrangements, skillfully used the connection observation method and the incomplete induction method to formally propose the conjecture of the distribution of Mersenne prime numbers (i.e. Zhou's conjecture) in 1992. . This conjecture deepens people's understanding of the properties of special prime numbers.

Mathematical conjectures are generally proposed through observation, verification, analogy, induction, generalization, etc. of a large number of facts. This method of going from the specific to the general and discovering uniqueness from individuality is an important driving force for mathematical research. The proposing and researching of mathematical conjectures vividly reflects the application of dialectics in mathematics and greatly promotes the research of mathematical methodology. In addition, mathematical conjectures often become an important indicator of the level of mathematical development: Fermat's conjecture produced algebraic number theory; Poincaré's conjecture helped people better study three-dimensional space; Goldbach's conjecture promoted the development of the sieve method and the circle method. Development, especially the discovery of almost prime numbers, exception sets, the three prime number theorem of small variables, etc.; the Riemann hypothesis proved the prime number theorem and the application of elliptic curve technology to encryption and decryption, digital signatures, key exchange, large number decomposition and prime number judgment etc.; the four-color problem was solved by electronic computers, thus opening up a new era of machine proof. In this sense, mathematical conjectures are not only "bright and gorgeous gems", but also "hens that can lay golden eggs."