**Abstract**

In this work we have studied the prime numbers in the model *<sup>P</sup>* <sup>¼</sup> *am* <sup>þ</sup> 1, *<sup>m</sup>*, *<sup>a</sup>*>1<sup>∈</sup> . and the number in the form *<sup>q</sup>* <sup>¼</sup> *ma<sup>m</sup>* <sup>þ</sup> *bm* <sup>þ</sup> 1 in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if *Cm*ð Þ¼ *<sup>a</sup> mam* <sup>þ</sup> 1 is a generalized Cullen number then *m<sup>a</sup><sup>m</sup>* � �ð Þ<sup>1</sup> *<sup>a</sup>* ð Þ *mod Cm*ð Þ *a* . In a second paper published in 2014, they also presented a test for Broth's numbers in Form *kpn* <sup>þ</sup> 1 where *<sup>k</sup>*<*p<sup>n</sup>*. These results are basically a generalization of the work of W. Bosma and H.C Williams who studied the cases, especially when *p* ¼ 2, 3, as well as a generalization of the primitive MillerRabin test. In this study in particular, we presented a test for numbers in the form *ma<sup>m</sup>* <sup>þ</sup> *bm* <sup>þ</sup> 1 in the form of a polynomial that highlights the properties of these numbers as well as a test for the Fermat and Mersinner numbers and *p* ¼ *ab* þ 1 *a*, *b*>1∈ and *p* ¼ *qa* þ 1 where *q is prime odd* are special cases of the number *ma<sup>m</sup>* <sup>þ</sup> *bm* <sup>þ</sup> 1 when *<sup>b</sup>* takes a specific value. For example, we proved if *<sup>p</sup>* <sup>¼</sup> *qa* <sup>þ</sup> 1 where q is odd prime and *<sup>a</sup>*>1<sup>∈</sup> where *<sup>π</sup><sup>j</sup>* <sup>¼</sup> <sup>1</sup> *q q j* � � then P*<sup>q</sup>*�<sup>2</sup> *<sup>j</sup>*¼<sup>1</sup> *<sup>π</sup>j*ð Þ �*Cm*ð Þ *<sup>a</sup> <sup>q</sup>*�*j*�<sup>1</sup> *<sup>q</sup>* � *am* ð Þ� *<sup>χ</sup>*ð Þ *<sup>m</sup>*,*q*�*am* ð Þ *mod p* Components of proof Binomial theorem Fermat's Litter Theorem Elementary algebra.

**Keywords:** broth numbers, Cullen number, polynomial, Fermat number, Mersinne numbers
