
New record prime: 2^{74,207,281}1 with 22,338,618 digits by Cooper, Woltman, Kurowski, Blosser & GIMPS (7 Jan 2016). 

Here we address the following frequently asked question. I have found a wonderful new result. What do I do with it?The answer is simple: publish it. There are a variety of ways to do this. You can send it to an email discussion list such as PrimesL, share it on a news list such as sci.math, or place it on a personal web page. However, the standard way to publish is to send your results to an electronic or print journal. By some estimates 20,000 new theorems a year are published in these journals. It is in journals that you see the lively and ongoing debate we call mathematics. It is by publishing in a journal, specifically a refereed journal, that you get the community's acceptance of your results. For these reasons, as well as for reasons of tradition, the bulk of mathematics is found in journalsvery little of current mathematics can be found on the web. So what journal do you submit to? Will they pay? Will your work get stolen? The way you get ready to write is to read. Find what others have said about your topic, especially what they have said recently. Look at the form in which they presented their work and the questions they ask and address. Look up the articles they quote. Only after you have read well, are you ready to write well. Next write your article and share the result with others. Mathematicians typically do this with friends and colleagues down the hall, then at mathematics meetings or through email and news groups. Sometimes they do it by emailing a preprint (finished but unpublished version of their paper) to others who have done similar work. Perhaps that person will have time to comment. Mathematicians share their work to gain from the experiences of others, to have others give opinions of our work: especially suggestion of what we might improve or incorporate into our paper. Most journals formalize this process by assigning one or more referees to submitted papers. The referee is an expert in the area and can see if the result is new, correct and significant enough to be worth publishing. Which journal do you submit to? At this point, you should be able to answer this question easily: submit it to one of the journals that has published related results recently. If you are not sure, return to step one and spend more time in the library reading, or online asking where you might read. It is very unlikely (nigh impossible) that your result is so new that there are no related results. Will you be paid? No. Journals do mathematicians a favor by being the vehicles for our dialog. In fact they sometimes ask larger institutions (not individuals) to "pay page set up fees" to help defer the cost of the publishing process. Will your work be stolen? No. If you have shared your results as suggested above, then your intellectual ownership is well established. Most mathematicians are honest, hard working, and have no desire to take another's work. However, if your work is good enough, they might quickly build on it and expand itin fact most of us hope to write something others want to use and build on. Other resources (from the American Mathematics Society or AMS, and the Mathematical Association of America or MAA): . 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 